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Title: A Short Account of the History of Mathematics Author: W. W. Rouse Ball Release Date: May 28, 2010 [EBook #31246] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICS ***

A SHORT ACCOUNT OF THE

HISTORY OF MATHEMATICS

BY

W. W. ROUSE BALL FELLOW OF TRINITY COLLEGE, CAMBRIDGE

DOVER PUBLICATIONS, INC. NEW YORK

This new Dover edition, first published in 1960, is an unabridged and unaltered republication of the author’s last revision—the fourth edition which appeared in 1908.

International Standard Book Number: 0-486-20630-0 Library of Congress Catalog Card Number: 60-3187

Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N. Y. 10014

Produced by Greg Lindahl, Viv, Juliet Sutherland, Nigel Blower and the Online Distributed Proofreading Team at http://www.pgdp.net

Transcriber’s Notes A small number of minor typographical errors and inconsistencies have been corrected. References to figures such as “on the next page” have been replaced with text such as “below” which is more suited to an eBook. Such changes are documented in the LATEX source: %[**TN: text of note]

PREFACE. The subject-matter of this book is a historical summary of the development of mathematics, illustrated by the lives and discoveries of those to whom the progress of the science is mainly due. It may serve as an introduction to more elaborate works on the subject, but primarily it is intended to give a short and popular account of those leading facts in the history of mathematics which many who are unwilling, or have not the time, to study it systematically may yet desire to know. The first edition was substantially a transcript of some lectures which I delivered in the year 1888 with the object of giving a sketch of the history, previous to the nineteenth century, that should be intelligible to any one acquainted with the elements of mathematics. In the second edition, issued in 1893, I rearranged parts of it, and introduced a good deal of additional matter. The scheme of arrangement will be gathered from the table of contents at the end of this preface. Shortly it is as follows. The first chapter contains a brief statement of what is known concerning the mathematics of the Egyptians and Phoenicians; this is introductory to the history of mathematics under Greek influence. The subsequent history is divided into three periods: first, that under Greek influence, chapters ii to vii; second, that of the middle ages and renaissance, chapters viii to xiii; and lastly that of modern times, chapters xiv to xix. In discussing the mathematics of these periods I have confined myself to giving the leading events in the history, and frequently have passed in silence over men or works whose influence was comparatively unimportant. Doubtless an exaggerated view of the discoveries of those mathematicians who are mentioned may be caused by the non-allusion to minor writers who preceded and prepared the way for them, but in all historical sketches this is to some extent inevitable, and I have done my best to guard against it by interpolating remarks on the progress

PREFACE

v

of the science at different times. Perhaps also I should here state that generally I have not referred to the results obtained by practical astronomers and physicists unless there was some mathematical interest in them. In quoting results I have commonly made use of modern notation; the reader must therefore recollect that, while the matter is the same as that of any writer to whom allusion is made, his proof is sometimes translated into a more convenient and familiar language. The greater part of my account is a compilation from existing histories or memoirs, as indeed must be necessarily the case where the works discussed are so numerous and cover so much ground. When authorities disagree I have generally stated only that view which seems to me to be the most probable; but if the question be one of importance, I believe that I have always indicated that there is a difference of opinion about it. I think that it is undesirable to overload a popular account with a mass of detailed references or the authority for every particular fact mentioned. For the history previous to 1758, I need only refer, once for all, to the closely printed pages of M. Cantor’s monumental Vorlesungen u ¨ber die Geschichte der Mathematik (hereafter alluded to as Cantor), which may be regarded as the standard treatise on the subject, but usually I have given references to the other leading authorities on which I have relied or with which I am acquainted. My account for the period subsequent to 1758 is generally based on the memoirs or monographs referred to in the footnotes, but the main facts to 1799 have been also enumerated in a supplementary volume issued by Prof. Cantor last year. I hope that my footnotes will supply the means of studying in detail the history of mathematics at any specified period should the reader desire to do so. My thanks are due to various friends and correspondents who have called my attention to points in the previous editions. I shall be grateful for notices of additions or corrections which may occur to any of my readers.

W. W. ROUSE BALL. TRINITY COLLEGE, CAMBRIDGE.

NOTE. The fourth edition was stereotyped in 1908, but no material changes have been made since the issue of the second edition in 1893, other duties having, for a few years, rendered it impossible for me to find time for any extensive revision. Such revision and incorporation of recent researches on the subject have now to be postponed till the cost of printing has fallen, though advantage has been taken of reprints to make trivial corrections and additions.

W. W. R. B. TRINITY COLLEGE, CAMBRIDGE.

vi

vii

TABLE OF CONTENTS. Preface . . . Table of Contents

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page iv vii

Chapter I. Egyptian and Phoenician Mathematics. The history of mathematics begins with that of the Ionian Greeks Greek indebtedness to Egyptians and Phoenicians . . . Knowledge of the science of numbers possessed by the Phoenicians Knowledge of the science of numbers possessed by the Egyptians Knowledge of the science of geometry possessed by the Egyptians Note on ignorance of mathematics shewn by the Chinese . .

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1 1 2 2 4 7

First Period. Mathematics under Greek Influence. This period begins with the teaching of Thales, circ. 600 b.c., and ends with the capture of Alexandria by the Mohammedans in or about 641 a.d. The characteristic feature of this period is the development of geometry.

Chapter II. The Ionian and Pythagorean Schools. Circ. 600 b.c.–400 b.c. Authorities . . . . . . . . The Ionian School . . . . . . . Thales, 640–550 b.c. . . . . . . His geometrical discoveries . . . . His astronomical teaching . . . . . Anaximander. Anaximenes. Mamercus. Mandryatus The Pythagorean School . . . . . . Pythagoras, 569–500 b.c. . . . . . The Pythagorean teaching . . . . . The Pythagorean geometry . . . .

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10 11 11 11 13 14 15 15 15 17

viii

TABLE OF CONTENTS The Pythagorean theory of numbers . . . . Epicharmus. Hippasus. Philolaus. Archippus. Lysis . . Archytas, circ. 400 b.c. . . . . . . . His solution of the duplication of a cube . . . Theodorus. Timaeus. Bryso . . . . . . Other Greek Mathematical Schools in the Fifth Century b.c. Oenopides of Chios . . . . . . . . Zeno of Elea. Democritus of Abdera . . . . .

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19 22 22 23 24 24 24 25

Chapter III. The Schools of Athens and Cyzicus. Circ. 420–300 b.c. Authorities . . . . . . . . . Mathematical teachers at Athens prior to 420 b.c. . . Anaxagoras. The Sophists. Hippias (The quadratrix). Antipho . . . . . . . . . Three problems in which these schools were specially interested Hippocrates of Chios, circ. 420 b.c. . . . . Letters used to describe geometrical diagrams . . Introduction in geometry of the method of reduction The quadrature of certain lunes . . . . . The problem of the duplication of the cube . . Plato, 429–348 b.c. . . . . . . . . Introduction in geometry of the method of analysis . Theorem on the duplication of the cube . . . Eudoxus, 408–355 b.c. . . . . . . . Theorems on the golden section . . . . . Introduction of the method of exhaustions . . Pupils of Plato and Eudoxus . . . . . . Menaechmus, circ. 340 b.c. . . . . . . Discussion of the conic sections . . . . . His two solutions of the duplication of the cube . Aristaeus. Theaetetus . . . . . . . Aristotle, 384–322 b.c. . . . . . . . Questions on mechanics. Letters used to indicate magnitudes

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27 27 27 29 30 31 31 32 32 34 34 35 36 36 36 37 38 38 38 38 39 39 40

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41 41 43 43 44 44 47

Chapter IV. The First Alexandrian School. Circ. 300–30 b.c. Authorities . . . . . . . . . Foundation of Alexandria . . . . . . . The Third Century before Christ . . . . . Euclid, circ. 330–275 b.c. . . . . . . Euclid’s Elements . . . . . . . The Elements as a text-book of geometry . . . The Elements as a text-book of the theory of numbers

. . . . . . .

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ix

TABLE OF CONTENTS Euclid’s other works . . . . . . . Aristarchus, circ. 310–250 b.c. . . . . . . Method of determining the distance of the sun . . Conon. Dositheus. Zeuxippus. Nicoteles . . . . Archimedes, 287–212 b.c. . . . . . . His works on plane geometry . . . . . His works on geometry of three dimensions . . His two papers on arithmetic, and the “cattle problem” His works on the statics of solids and fluids . . His astronomy . . . . . . . . The principles of geometry assumed by Archimedes . Apollonius, circ. 260–200 b.c. . . . . . His conic sections . . . . . . . His other works . . . . . . . . His solution of the duplication of the cube . . Contrast between his geometry and that of Archimedes Eratosthenes, 275–194 b.c. . . . . . . The Sieve of Eratosthenes . . . . . . The Second Century before Christ . . . . . Hypsicles (Euclid, book xiv). Nicomedes. Diocles . . Perseus. Zenodorus . . . . . . . . Hipparchus, circ. 130 b.c. . . . . . . Foundation of scientific astronomy . . . . Foundation of trigonometry . . . . . Hero of Alexandria, circ. 125 b.c. . . . . . Foundation of scientific engineering and of land-surveying Area of a triangle determined in terms of its sides . Features of Hero’s works . . . . . . The First Century before Christ . . . . . Theodosius . . . . . . . . . Dionysodorus . . . . . . . . . End of the First Alexandrian School . . . . . Egypt constituted a Roman province . . . .

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49 51 51 52 53 55 58 59 60 63 63 63 64 66 67 68 69 69 70 70 71 71 72 73 73 73 74 75 76 76 76 76 76

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78 79 79 79 79 80 80 80 80 80

Chapter V. The Second Alexandrian School. 30 b.c.–641 a.d. Authorities . . . . . . . . . The First Century after Christ . . . . . . Serenus. Menelaus . . . . . . . . Nicomachus . . . . . . . . . Introduction of the arithmetic current in medieval Europe The Second Century after Christ . . . . . Theon of Smyrna. Thymaridas . . . . . . Ptolemy, died in 168 . . . . . . . The Almagest . . . . . . . . Ptolemy’s astronomy . . . . . . .

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x

TABLE OF CONTENTS Ptolemy’s geometry . . . . . . . The Third Century after Christ . . . . . Pappus, circ. 280 . . . . . . . . The Συναγωγή, a synopsis of Greek mathematics . The Fourth Century after Christ . . . . . Metrodorus. Elementary problems in arithmetic and algebra Three stages in the development of algebra . . . Diophantus, circ. 320 (?) . . . . . . Introduction of syncopated algebra in his Arithmetic The notation, methods, and subject-matter of the work His Porisms . . . . . . . . Subsequent neglect of his discoveries . . . . Iamblichus . . . . . . . . . Theon of Alexandria. Hypatia . . . . . . Hostility of the Eastern Church to Greek science . . The Athenian School (in the Fifth Century) . . . Proclus, 412–485. Damascius. Eutocius . . . . Roman Mathematics . . . . . . . . Nature and extent of the mathematics read at Rome . Contrast between the conditions at Rome and at Alexandria End of the Second Alexandrian School . . . . The capture of Alexandria, and end of the Alexandrian Schools

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82 83 83 83 85 85 86 86 87 87 91 92 92 92 93 93 93 94 94 95 96 96

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97 97 99 100

Chapter VI. The Byzantine School. 641–1453. Preservation of works of the great Greek Mathematicians . Hero of Constantinople. Psellus. Planudes. Barlaam. Argyrus . Nicholas Rhabdas, Pachymeres. Moschopulus (Magic Squares) Capture of Constantinople, and dispersal of Greek Mathematicians

. . . .

Chapter VII. Systems of Numeration and Primitive Arithmetic. Authorities . . . . . . . . . . Methods of counting and indicating numbers among primitive races Use of the abacus or swan-pan for practical calculation . . Methods of representing numbers in writing . . . . The Roman and Attic symbols for numbers . . . . The Alexandrian (or later Greek) symbols for numbers . . Greek arithmetic . . . . . . . . . Adoption of the Arabic system of notation among civilized races

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101 101 103 105 105 106 106 107

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TABLE OF CONTENTS

Second Period. Mathematics of the Middle Ages and of the Renaissance. This period begins about the sixth century, and may be said to end with the invention of analytical geometry and of the infinitesimal calculus. The characteristic feature of this period is the creation or development of modern arithmetic, algebra, and trigonometry.

Chapter VIII. The Rise Of Learning In Western Europe. Circ. 600–1200. Authorities . . . . . . . . Education in the Sixth, Seventh, and Eighth Centuries The Monastic Schools . . . . . . Boethius, circ. 475–526 . . . . . . Medieval text-books in geometry and arithmetic Cassiodorus, 490–566. Isidorus of Seville, 570–636 . The Cathedral and Conventual Schools . . . The Schools of Charles the Great . . . . Alcuin, 735–804 . . . . . . . Education in the Ninth and Tenth Centuries . . Gerbert (Sylvester II.), died in 1003 . . . . Bernelinus . . . . . . . . The Early Medieval Universities . . . . Rise during the twelfth century of the earliest universities Development of the medieval universities . . . Outline of the course of studies in a medieval university

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109 109 109 110 110 111 111 111 111 113 113 115 115 115 116 117

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120 120 121 121 122 122 123 123 125 125 127 129 129 130 130

Chapter IX. The Mathematics Of The Arabs. Authorities . . . . . . . . Extent of Mathematics obtained from Greek Sources . The College of Scribes . . . . . . Extent of Mathematics obtained from the (Aryan) Hindoos Arya-Bhata, circ. 530 . . . . . . His algebra and trigonometry (in his Aryabhathiya) Brahmagupta, circ. 640 . . . . . . His algebra and geometry (in his Siddhanta) . Bhaskara, circ. 1140 . . . . . . The Lilavati or arithmetic; decimal numeration used The Bija Ganita or algebra . . . . Development of Mathematics in Arabia . . . ¯ rizm¯i, circ. 830 . Alkarismi or Al-Khwa . . His Al-gebr we’ l mukabala . . . . His solution of a quadratic equation . . .

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xii

TABLE OF CONTENTS Introduction of Arabic or Indian system of numeration Tabit ibn Korra, 836–901; solution of a cubic equation Alkayami. Alkarki. Development of algebra . . . Albategni. Albuzjani. Development of trigonometry . . Alhazen. Abd-al-gehl. Development of geometry . . Characteristics of the Arabian School . . . .

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131 132 132 133 134 134

Chapter X. Introduction of Arabian Works into Europe. Circ. 1150–1450. The Eleventh Century . . . . . . . Moorish Teachers. Geber ibn Aphla. Arzachel . . . The Twelfth Century . . . . . . . Adelhard of Bath . . . . . . . . Ben-Ezra. Gerard. John Hispalensis . . . . . The Thirteenth Century . . . . . . . Leonardo of Pisa, circ. 1175–1230 . . . . The Liber Abaci, 1202 . . . . . . The introduction of the Arabic numerals into commerce The introduction of the Arabic numerals into science The mathematical tournament . . . . . Frederick II., 1194–1250 . . . . . . . Jordanus, circ. 1220 . . . . . . . His De Numeris Datis; syncopated algebra . . Holywood . . . . . . . . . . Roger Bacon, 1214–1294 . . . . . . Campanus . . . . . . . . . The Fourteenth Century . . . . . . . Bradwardine . . . . . . . . . Oresmus . . . . . . . . . . The reform of the university curriculum . . . . The Fifteenth Century . . . . . . . Beldomandi . . . . . . . . .

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136 136 137 137 137 138 138 138 139 139 140 141 141 142 144 144 147 147 147 147 148 149 149

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151 151 151 152 154 156 156 162

Chapter XI. The Development Of Arithmetic. Circ. 1300–1637. Authorities . . . . . . . . . The Boethian arithmetic . . . . . . . Algorism or modern arithmetic . . . . . . The Arabic (or Indian) symbols: history of . . . Introduction into Europe by science, commerce, and calendars Improvements introduced in algoristic arithmetic . . (i) Simplification of the fundamental processes . . (ii) Introduction of signs for addition and subtraction

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xiii

TABLE OF CONTENTS (iii) Invention of logarithms, 1614 (iv) Use of decimals, 1619 . .

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162 163

Chapter XII. The Mathematics of the Renaissance. Circ. 1450–1637. Authorities . . . . . . . . . Effect of invention of printing. The renaissance . . Development of Syncopated Algebra and Trigonometry . Regiomontanus, 1436–1476 . . . . . . His De Triangulis (printed in 1496) . . . . Purbach, 1423–1461. Cusa, 1401–1464. Chuquet, circ. 1484 Introduction and origin of symbols + and − . . . Pacioli or Lucas di Burgo, circ. 1500 . . . . His arithmetic and geometry, 1494 . . . . Leonardo da Vinci, 1452–1519 . . . . . . D¨ urer, 1471–1528. Copernicus, 1473–1543 . . . Record, 1510–1558; introduction of symbol for equality . Rudolff, circ. 1525. Riese, 1489–1559 . . . . Stifel, 1486–1567 . . . . . . . . His Arithmetica Integra, 1544 . . . . . Tartaglia, 1500–1557 . . . . . . . His solution of a cubic equation, 1535 . . . His arithmetic, 1556–1560 . . . . . . Cardan, 1501–1576 . . . . . . . . His Ars Magna, 1545; the third work printed on algebra. His solution of a cubic equation . . . . . Ferrari, 1522–1565; solution of a biquadratic equation . Rheticus, 1514–1576. Maurolycus. Borrel. Xylander . Commandino. Peletier. Romanus. Pitiscus. Ramus. 1515–1572 Bombelli, circ. 1570 . . . . . . . . Development of Symbolic Algebra . . . . . Vieta, 1540–1603 . . . . . . . . The In Artem; introduction of symbolic algebra, 1591 Vieta’s other works . . . . . . . Girard, 1595–1632; development of trigonometry and algebra Napier, 1550–1617; introduction of logarithms, 1614 . Briggs, 1561–1631; calculations of tables of logarithms . Harriot, 1560–1621; development of analysis in algebra . Oughtred, 1574–1660 . . . . . . . The Origin of the more Common Symbols in Algebra .

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165 165 166 166 167 170 171 173 173 176 176 177 178 178 179 180 181 182 183 184 186 186 187 187 188 189 189 191 192 194 195 196 196 197 198

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TABLE OF CONTENTS

Chapter XIII. The Close of the Renaissance. Circ. 1586–1637. Authorities . . . . . . . . . Development of Mechanics and Experimental Methods . Stevinus, 1548–1620 . . . . . . . Commencement of the modern treatment of statics, 1586 Galileo, 1564–1642 . . . . . . . . Commencement of the science of dynamics . . Galileo’s astronomy . . . . . . . Francis Bacon, 1561–1626. Guldinus, 1577–1643 . . Wright, 1560–1615; construction of maps . . . . Snell, 1591–1626 . . . . . . . . Revival of Interest in Pure Geometry . . . . Kepler, 1571–1630 . . . . . . . . His Paralipomena, 1604; principle of continuity . . His Stereometria, 1615; use of infinitesimals . . Kepler’s laws of planetary motion, 1609 and 1619 . Desargues, 1593–1662 . . . . . . . His Brouillon project; use of projective geometry . Mathematical Knowledge at the Close of the Renaissance .

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202 202 202 203 205 205 206 208 209 210 210 210 211 212 212 213 213 214

Third period. Modern Mathematics. This period begins with the invention of analytical geometry and the infinitesimal calculus. The mathematics is far more complex than that produced in either of the preceding periods: but it may be generally described as characterized by the development of analysis, and its application to the phenomena of nature.

Chapter XIV. The History of Modern Mathematics. Treatment of the subject . . . . . . . . Invention of analytical geometry and the method of indivisibles Invention of the calculus . . . . . . . . Development of mechanics . . . . . . . Application of mathematics to physics . . . . . Recent development of pure mathematics . . . . .

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217 218 218 219 219 220

Chapter XV. History of Mathematics from Descartes to Huygens. Circ. 1635–1675. Authorities . . . Descartes, 1596–1650 . His views on philosophy

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221 221 224

xv

TABLE OF CONTENTS His invention of analytical geometry, 1637 . . . . . His algebra, optics, and theory of vortices . . . . . Cavalieri, 1598–1647 . . . . . . . . . The method of indivisibles . . . . . . . . Pascal, 1623–1662 . . . . . . . . . . His geometrical conics . . . . . . . . The arithmetical triangle . . . . . . . . Foundation of the theory of probabilities, 1654 . . . . His discussion of the cycloid . . . . . . . Wallis, 1616–1703 . . . . . . . . . . The Arithmetica Infinitorum, 1656 . . . . . . Law of indices in algebra . . . . . . . . Use of series in quadratures . . . . . . . Earliest rectification of curves, 1657 . . . . . . Wallis’s algebra . . . . . . . . . . Fermat, 1601–1665 . . . . . . . . . . His investigations on the theory of numbers . . . . His use in geometry of analysis and of infinitesimals . . . Foundation of the theory of probabilities, 1654 . . . . Huygens, 1629–1695 . . . . . . . . . The Horologium Oscillatorium, 1673 . . . . . . The undulatory theory of light . . . . . . . Other Mathematicians of this Time . . . . . . . Bachet . . . . . . . . . . . . Mersenne; theorem on primes and perfect numbers . . . . Roberval. Van Schooten. Saint-Vincent . . . . . . Torricelli. Hudde. Fr´enicle . . . . . . . . De Laloub`ere. Mercator. Barrow; the differential triangle . . Brouncker; continued fractions . . . . . . . . James Gregory; distinction between convergent and divergent series Sir Christopher Wren . . . . . . . . . Hooke. Collins . . . . . . . . . . . Pell. Sluze. Viviani . . . . . . . . . . Tschirnhausen. De la Hire. Roemer. Rolle . . . . .

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224 227 229 230 232 234 234 235 236 237 238 238 239 240 241 241 242 246 247 248 249 250 251 252 252 253 254 254 257 258 259 259 260 261

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263 263 264 265 266 267 267 268 269 272

Chapter XVI. The Life and Works of Newton. Authorities . . . . . . Newton’s school and undergraduate life . Investigations in 1665–1666 on fluxions, optics, His views on gravitation, 1666 . . Researches in 1667–1669 . . . . Elected Lucasian professor, 1669 . . Optical lectures and discoveries, 1669–1671 Emission theory of light, 1675 . . . The Leibnitz Letters, 1676 . . . Discoveries on gravitation, 1679 . .

. . . . . . . . and gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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TABLE OF CONTENTS Discoveries and lectures on algebra, 1673–1683 . . Discoveries and lectures on gravitation, 1684 . . The Principia, 1685–1686 . . . . . . The subject-matter of the Principia . . . Publication of the Principia . . . . Investigations and work from 1686 to 1696 . . Appointment at the Mint, and removal to London, 1696 Publication of the Optics, 1704 . . . . . Appendix on classification of cubic curves . . Appendix on quadrature by means of infinite series Appendix on method of fluxions . . . The invention of fluxions and the infinitesimal calculus Newton’s death, 1727 . . . . . . List of his works . . . . . . . Newton’s character . . . . . . . Newton’s discoveries . . . . . . .

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272 274 275 276 278 278 279 279 279 281 282 286 286 286 287 289

Chapter XVII. Leibnitz and the Mathematicians of the First Half of the Eighteenth Century. Authorities . . . . . . . . . . Leibnitz and the Bernoullis . . . . . . . Leibnitz, 1646–1716 . . . . . . . . His system of philosophy, and services to literature . . The controversy as to the origin of the calculus . . . His memoirs on the infinitesimal calculus . . . . His papers on various mechanical problems . . . Characteristics of his work . . . . . . . James Bernoulli, 1654–1705 . . . . . . . John Bernoulli, 1667–1748 . . . . . . . The younger Bernouillis . . . . . . . . Development of Analysis on the Continent . . . . L’Hospital, 1661–1704 . . . . . . . . Varignon, 1654–1722. De Montmort. Nicole . . . . Parent. Saurin. De Gua. Cramer, 1704–1752 . . . . Riccati, 1676–1754. Fagnano, 1682–1766 . . . . . Clairaut, 1713–1765 . . . . . . . . D’Alembert, 1717–1783 . . . . . . . . Solution of a partial differential equation of the second order Daniel Bernoulli, 1700–1782 . . . . . . . English Mathematicians of the Eighteenth Century . . . David Gregory, 1661–1708. Halley, 1656–1742 . . . . Ditton, 1675–1715 . . . . . . . . . Brook Taylor, 1685–1731 . . . . . . . Taylor’s theorem . . . . . . . . Taylor’s physical researches . . . . . . Cotes, 1682–1716 . . . . . . . . .

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291 291 291 293 293 298 299 301 301 302 303 304 304 305 305 306 307 308 309 311 312 312 313 313 314 314 315

xvii

TABLE OF CONTENTS Demoivre, 1667–1754; development of trigonometry Maclaurin, 1698–1746 . . . . . His geometrical discoveries . . . The Treatise of Fluxions . . . . His propositions on attractions . . . Stewart, 1717–1785. Thomas Simpson, 1710–1761

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315 316 317 318 318 319

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322 323 323 324 326 326 326 327 329 330 330 331 334 337 338 338 339 339 340 340 341 341 342 343 344 344 345 346 346 347 348 348 349 349 350 350 351

Chapter XVIII. Lagrange, Laplace, and their Contemporaries. Circ. 1740–1830. Characteristics of the mathematics of the period . . . Development of Analysis and Mechanics . . . . . Euler, 1707–1783 . . . . . . . . . The Introductio in Analysin Infinitorum, 1748 . . . The Institutiones Calculi Differentialis, 1755 . . . The Institutiones Calculi Integralis, 1768–1770 . . . The Anleitung zur Algebra, 1770 . . . . . Euler’s works on mechanics and astronomy . . . Lambert, 1728–1777 . . . . . . . . . B´ezout, 1730–1783. Trembley, 1749–1811. Arbogast, 1759–1803 Lagrange, 1736–1813 . . . . . . . . Memoirs on various subjects . . . . . . The M´ecanique analytique, 1788 . . . . . The Th´eorie and Calcul des fonctions, 1797, 1804 . . The R´esolution des ´equations num´eriques, 1798. . . Characteristics of Lagrange’s work . . . . . Laplace, 1749–1827 . . . . . . . . Memoirs on astronomy and attractions, 1773–1784 . . Use of spherical harmonics and the potential . . . Memoirs on problems in astronomy, 1784–1786 . . . The M´ecanique c´eleste and Exposition du syst`eme du monde The Nebular Hypothesis . . . . . . . The Meteoric Hypothesis . . . . . . . The Th´eorie analytique des probabilit´es, 1812 . . . The Method of Least Squares . . . . . . Other researches in pure mathematics and in physics . Characteristics of Laplace’s work . . . . . Character of Laplace . . . . . . . . Legendre, 1752–1833 . . . . . . . . His memoirs on attractions . . . . . . The Th´eorie des nombres, 1798 . . . . . . Law of quadratic reciprocity . . . . . . The Calcul int´egral and the Fonctions elliptiques . . Pfaff, 1765–1825 . . . . . . . . . Creation of Modern Geometry . . . . . . . Monge, 1746–1818 . . . . . . . . . Lazare Carnot, 1753–1823. Poncelet, 1788–1867 . . .

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xviii

TABLE OF CONTENTS Development of Mathematical Physics . . . Cavendish, 1731–1810 . . . . . . Rumford, 1753–1815. Young, 1773–1829 . . . Dalton, 1766–1844 . . . . . . . Fourier, 1768–1830 . . . . . . . Sadi Carnot; foundation of thermodynamics . . Poisson, 1781–1840 . . . . . . . Amp`ere, 1775–1836. Fresnel, 1788–1827. Biot, 1774–1862 Arago, 1786–1853 . . . . . . . Introduction of Analysis into England . . . Ivory, 1765–1842 . . . . . . . The Cambridge Analytical School . . . . Woodhouse, 1773–1827 . . . . . . Peacock, 1791–1858. Babbage, 1792–1871. John Herschel,

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353 353 353 354 355 356 356 358 359 360 360 361 361 362

Chapter XIX. Mathematics of the Nineteenth Century. Creation of new branches of mathematics . . . . . Difficulty in discussing the mathematics of this century . . Account of contemporary work not intended to be exhaustive . Authorities . . . . . . . . . . Gauss, 1777–1855 . . . . . . . . . Investigations in astronomy . . . . . . Investigations in electricity . . . . . . The Disquisitiones Arithmeticae, 1801 . . . . His other discoveries . . . . . . . . Comparison of Lagrange, Laplace, and Gauss . . . Dirichlet, 1805–1859 . . . . . . . . . Development of the Theory of Numbers . . . . . Eisenstein, 1823–1852 . . . . . . . . Henry Smith, 1826–1883 . . . . . . . . Kummer, 1810–1893 . . . . . . . . . Notes on other writers on the Theory of Numbers . . . Development of the Theory of Functions of Multiple Periodicity Abel, 1802–1829. Abel’s Theorem . . . . . . Jacobi, 1804–1851 . . . . . . . . . Riemann, 1826–1866 . . . . . . . . Notes on other writers on Elliptic and Abelian Functions . . Weierstrass, 1815–1897 . . . . . . . Notes on recent writers on Elliptic and Abelian Functions . The Theory of Functions . . . . . . . . Development of Higher Algebra . . . . . . Cauchy, 1789–1857 . . . . . . . . . Argand, 1768–1822; geometrical interpretation of complex numbers Sir William Hamilton, 1805–1865; introduction of quaternions Grassmann, 1809–1877; his non-commutative algebra, 1844 . Boole, 1815–1864. De Morgan, 1806–1871 . . . .

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365 365 365 366 367 368 369 371 372 373 373 374 374 374 377 377 378 379 380 381 382 382 383 384 385 385 387 387 389 389

xix

TABLE OF CONTENTS Galois, 1811–1832; theory of discontinuous substitution groups Cayley, 1821–1895 . . . . . . . . . Sylvester, 1814–1897 . . . . . . . . Lie, 1842–1889; theory of continuous substitution groups . . Hermite, 1822–1901 . . . . . . . . Notes on other writers on Higher Algebra . . . . . Development of Analytical Geometry . . . . . Notes on some recent writers on Analytical Geometry . . Line Geometry . . . . . . . . . . Analysis. Names of some recent writers on Analysis . . . Development of Synthetic Geometry . . . . . . Steiner, 1796–1863 . . . . . . . . . Von Staudt, 1798–1867 . . . . . . . . Other writers on modern Synthetic Geometry . . . . Development of Non-Euclidean Geometry . . . . . Euclid’s Postulate on Parallel Lines . . . . . Hyperbolic Geometry. Elliptic Geometry . . . . Congruent Figures . . . . . . . . Foundations of Mathematics. Assumptions made in the subject Kinematics . . . . . . . . . . Development of the Theory of Mechanics, treated Graphically . Development of Theoretical Mechanics, treated Analytically . Notes on recent writers on Mechanics . . . . . Development of Theoretical Astronomy . . . . . Bessel, 1784–1846 . . . . . . . . . Leverrier, 1811–1877. Adams, 1819–1892 . . . . . Notes on other writers on Theoretical Astronomy . . . Recent Developments . . . . . . . . Development of Mathematical Physics . . . . .

Index

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390 390 391 392 392 393 395 395 396 396 397 397 398 398 398 399 399 401 402 402 402 403 405 405 405 406 407 408 409

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410

1

CHAPTER I. egyptian and phoenician mathematics. The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks. The subsequent history may be divided into three periods, the distinctions between which are tolerably well marked. The first period is that of the history of mathematics under Greek influence, this is discussed in chapters ii to vii; the second is that of the mathematics of the middle ages and the renaissance, this is discussed in chapters viii to xiii; the third is that of modern mathematics, and this is discussed in chapters xiv to xix. Although the history of mathematics commences with that of the Ionian schools, there is no doubt that those Greeks who first paid attention to the subject were largely indebted to the previous investigations of the Egyptians and Phoenicians. Our knowledge of the mathematical attainments of those races is imperfect and partly conjectural, but, such as it is, it is here briefly summarised. The definite history begins with the next chapter. On the subject of prehistoric mathematics, we may observe in the first place that, though all early races which have left records behind them knew something of numeration and mechanics, and though the majority were also acquainted with the elements of land-surveying, yet the rules which they possessed were in general founded only on the results of observation and experiment, and were neither deduced from nor did they form part of any science. The fact then that various nations in the vicinity of Greece had reached a high state of civilisation does not justify us in assuming that they had studied mathematics. The only races with whom the Greeks of Asia Minor (amongst whom our history begins) were likely to have come into frequent contact were those inhabiting the eastern littoral of the Mediterranean; and Greek

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tradition uniformly assigned the special development of geometry to the Egyptians, and that of the science of numbers either to the Egyptians or to the Phoenicians. I discuss these subjects separately. First, as to the science of numbers. So far as the acquirements of the Phoenicians on this subject are concerned it is impossible to speak with certainty. The magnitude of the commercial transactions of Tyre and Sidon necessitated a considerable development of arithmetic, to which it is probable the name of science might be properly applied. A Babylonian table of the numerical value of the squares of a series of consecutive integers has been found, and this would seem to indicate that properties of numbers were studied. According to Strabo the Tyrians paid particular attention to the sciences of numbers, navigation, and astronomy; they had, we know, considerable commerce with their neighbours and kinsmen the Chaldaeans; and B¨ockh says that they regularly supplied the weights and measures used in Babylon. Now the Chaldaeans had certainly paid some attention to arithmetic and geometry, as is shown by their astronomical calculations; and, whatever was the extent of their attainments in arithmetic, it is almost certain that the Phoenicians were equally proficient, while it is likely that the knowledge of the latter, such as it was, was communicated to the Greeks. On the whole it seems probable that the early Greeks were largely indebted to the Phoenicians for their knowledge of practical arithmetic or the art of calculation, and perhaps also learnt from them a few properties of numbers. It may be worthy of note that Pythagoras was a Phoenician; and according to Herodotus, but this is more doubtful, Thales was also of that race. I may mention that the almost universal use of the abacus or swanpan rendered it easy for the ancients to add and subtract without any knowledge of theoretical arithmetic. These instruments will be described later in chapter vii; it will be sufficient here to say that they afford a concrete way of representing a number in the decimal scale, and enable the results of addition and subtraction to be obtained by a merely mechanical process. This, coupled with a means of representing the result in writing, was all that was required for practical purposes. We are able to speak with more certainty on the arithmetic of the Egyptians. About forty years ago a hieratic papyrus,1 forming part 1

See Ein mathematisches Handbuch der alten Aegypter, by A. Eisenlohr, second edition, Leipzig, 1891; see also Cantor, chap. i; and A Short History of Greek Mathematics, by J. Gow, Cambridge, 1884, arts. 12–14. Besides these authorities the

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of the Rhind collection in the British Museum, was deciphered, which has thrown considerable light on their mathematical attainments. The manuscript was written by a scribe named Ahmes at a date, according to Egyptologists, considerably more than a thousand years before Christ, and it is believed to be itself a copy, with emendations, of a treatise more than a thousand years older. The work is called “directions for knowing all dark things,” and consists of a collection of problems in arithmetic and geometry; the answers are given, but in general not the processes by which they are obtained. It appears to be a summary of rules and questions familiar to the priests. The first part deals with the reduction of fractions of the form 2/(2n + 1) to a sum of fractions each of whose numerators is unity: 1 1 1 1 2 is the sum of 24 , 58 , 174 , and 232 ; for example, Ahmes states that 29 1 1 1 2 and 97 is the sum of 56 , 679 , and 776 . In all the examples n is less than 50. Probably he had no rule for forming the component fractions, and the answers given represent the accumulated experiences of previous writers: in one solitary case, however, he has indicated his method, for, after having asserted that 23 is the sum of 12 and 16 , he adds that therefore two-thirds of one-fifth is equal to the sum of a half of a fifth 1 1 and a sixth of a fifth, that is, to 10 + 30 . That so much attention was paid to fractions is explained by the fact that in early times their treatment was found difficult. The Egyptians and Greeks simplified the problem by reducing a fraction to the sum of several fractions, in each of which the numerator was unity, the sole exception to this rule being the fraction 23 . This remained the Greek practice until the sixth century of our era. The Romans, on the other hand, generally kept the denominator constant and equal to twelve, expressing the fraction (approximately) as so many twelfths. The Babylonians did the same in astronomy, except that they used sixty as the constant denominator; and from them through the Greeks the modern division of a degree into sixty equal parts is derived. Thus in one way or the other the difficulty of having to consider changes in both numerator and denominator was evaded. To-day when using decimals we often keep a fixed denominator, thus reverting to the Roman practice. After considering fractions Ahmes proceeds to some examples of the fundamental processes of arithmetic. In multiplication he seems to have papyrus has been discussed in memoirs by L. Rodet, A. Favaro, V. Bobynin, and E. Weyr.

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relied on repeated additions. Thus in one numerical example, where he requires to multiply a certain number, say a, by 13, he first multiplies by 2 and gets 2a, then he doubles the results and gets 4a, then he again doubles the result and gets 8a, and lastly he adds together a, 4a, and 8a. Probably division was also performed by repeated subtractions, but, as he rarely explains the process by which he arrived at a result, this is not certain. After these examples Ahmes goes on to the solution of some simple numerical equations. For example, he says “heap, its seventh, its whole, it makes nineteen,” by which he means that the object is to find a number such that the sum of it and one-seventh of it shall be together equal to 19; and he gives as the answer 16 + 12 + 18 , which is correct. The arithmetical part of the papyrus indicates that he had some idea of algebraic symbols. The unknown quantity is always represented by the symbol which means a heap; addition is sometimes represented by a pair of legs walking forwards, subtraction by a pair of legs walking −. backwards or by a flight of arrows; and equality by the sign < The latter part of the book contains various geometrical problems to which I allude later. He concludes the work with some arithmeticoalgebraical questions, two of which deal with arithmetical progressions and seem to indicate that he knew how to sum such series. Second, as to the science of geometry. Geometry is supposed to have had its origin in land-surveying; but while it is difficult to say when the study of numbers and calculation—some knowledge of which is essential in any civilised state—became a science, it is comparatively easy to distinguish between the abstract reasonings of geometry and the practical rules of the land-surveyor. Some methods of land-surveying must have been practised from very early times, but the universal tradition of antiquity asserted that the origin of geometry was to be sought in Egypt. That it was not indigenous to Greece, and that it arose from the necessity of surveying, is rendered the more probable by the derivation of the word from γ˜ η, the earth, and μετρέω, I measure. Now the Greek geometricians, as far as we can judge by their extant works, always dealt with the science as an abstract one: they sought for theorems which should be absolutely true, and, at any rate in historical times, would have argued that to measure quantities in terms of a unit which might have been incommensurable with some of the magnitudes considered would have made their results mere approximations to the truth. The name does not therefore refer to their practice. It is not, however, unlikely that it indicates the use which was made of geome-

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try among the Egyptians from whom the Greeks learned it. This also agrees with the Greek traditions, which in themselves appear probable; for Herodotus states that the periodical inundations of the Nile (which swept away the landmarks in the valley of the river, and by altering its course increased or decreased the taxable value of the adjoining lands) rendered a tolerably accurate system of surveying indispensable, and thus led to a systematic study of the subject by the priests. We have no reason to think that any special attention was paid to geometry by the Phoenicians, or other neighbours of the Egyptians. A small piece of evidence which tends to show that the Jews had not paid much attention to it is to be found in the mistake made in their sacred books,1 where it is stated that the circumference of a circle is three times its diameter: the Babylonians2 also reckoned that π was equal to 3. Assuming, then, that a knowledge of geometry was first derived by the Greeks from Egypt, we must next discuss the range and nature of Egyptian geometry.3 That some geometrical results were known at a date anterior to Ahmes’s work seems clear if we admit, as we have reason to do, that, centuries before it was written, the following method of obtaining a right angle was used in laying out the groundplan of certain buildings. The Egyptians were very particular about the exact orientation of their temples; and they had therefore to obtain with accuracy a north and south line, as also an east and west line. By observing the points on the horizon where a star rose and set, and taking a plane midway between them, they could obtain a north and south line. To get an east and west line, which had to be drawn at right angles to this, certain professional “rope-fasteners” were employed. These men used a rope ABCD divided by knots or marks at B and C, so that the lengths AB, BC, CD were in the ratio 3 : 4 : 5. The length BC was placed along the north and south line, and pegs P and Q inserted at the knots B and C. The piece BA (keeping it stretched all the time) was then rotated round the peg P , and similarly the piece CD was rotated round the peg Q, until the ends A and D coincided; the point thus indicated was marked by a peg R. The result was to form a triangle P QR whose sides RP , P Q, QR were in the ratio 3 : 4 : 5. The angle of 1

I. Kings, chap. vii, verse 23, and II. Chronicles, chap. iv, verse 2. See J. Oppert, Journal Asiatique, August 1872, and October 1874. 3 See Eisenlohr; Cantor, chap. ii; Gow, arts. 75, 76; and Die Geometrie der alten Aegypter, by E. Weyr, Vienna, 1884. 2

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the triangle at P would then be a right angle, and the line P R would give an east and west line. A similar method is constantly used at the present time by practical engineers for measuring a right angle. The property employed can be deduced as a particular case of Euc. i, 48; and there is reason to think that the Egyptians were acquainted with the results of this proposition and of Euc. i, 47, for triangles whose sides are in the ratio mentioned above. They must also, there is little doubt, have known that the latter proposition was true for an isosceles right-angled triangle, as this is obvious if a floor be paved with tiles of that shape. But though these are interesting facts in the history of the Egyptian arts we must not press them too far as showing that geometry was then studied as a science. Our real knowledge of the nature of Egyptian geometry depends mainly on the Rhind papyrus. Ahmes commences that part of his papyrus which deals with geometry by giving some numerical instances of the contents of barns. Unluckily we do not know what was the usual shape of an Egyptian barn, but where it is defined by three linear measurements, say a, b, and c, the answer is always given as if he had formed the expression a × b × (c + 12 c). He next proceeds to find the areas of certain rectilineal figures; if the text be correctly interpreted, some of these results are wrong. He then goes on to find the area of a circular field of diameter 12—no unit of length being mentioned—and gives the result as (d − 19 d)2 , where d is the diameter of the circle: this is equivalent to taking 3.1604 as the value of π, the actual value being very approximately 3.1416. Lastly, Ahmes gives some problems on pyramids. These long proved incapable of interpretation, but Cantor and Eisenlohr have shown that Ahmes was attempting to find, by means of data obtained from the measurement of the external dimensions of a building, the ratio of certain other dimensions which could not be directly measured: his process is equivalent to determining the trigonometrical ratios of certain angles. The data and the results given agree closely with the dimensions of some of the existing pyramids. Perhaps all Ahmes’s geometrical results were intended only as approximations correct enough for practical purposes. It is noticeable that all the specimens of Egyptian geometry which we possess deal only with particular numerical problems and not with general theorems; and even if a result be stated as universally true, it was probably proved to be so only by a wide induction. We shall see later that Greek geometry was from its commencement deductive. There are reasons for thinking that Egyptian geometry and arithmetic

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made little or no progress subsequent to the date of Ahmes’s work; and though for nearly two hundred years after the time of Thales Egypt was recognised by the Greeks as an important school of mathematics, it would seem that, almost from the foundation of the Ionian school, the Greeks outstripped their former teachers. It may be added that Ahmes’s book gives us much that idea of Egyptian mathematics which we should have gathered from statements about it by various Greek and Latin authors, who lived centuries later. Previous to its translation it was commonly thought that these statements exaggerated the acquirements of the Egyptians, and its discovery must increase the weight to be attached to the testimony of these authorities. We know nothing of the applied mathematics (if there were any) of the Egyptians or Phoenicians. The astronomical attainments of the Egyptians and Chaldaeans were no doubt considerable, though they were chiefly the results of observation: the Phoenicians are said to have confined themselves to studying what was required for navigation. Astronomy, however, lies outside the range of this book. I do not like to conclude the chapter without a brief mention of the Chinese, since at one time it was asserted that they were familiar with the sciences of arithmetic, geometry, mechanics, optics, navigation, and astronomy nearly three thousand years ago, and a few writers were inclined to suspect (for no evidence was forthcoming) that some knowledge of this learning had filtered across Asia to the West. It is true that at a very early period the Chinese were acquainted with several geometrical or rather architectural implements, such as the rule, square, compasses, and level; with a few mechanical machines, such as the wheel and axle; that they knew of the characteristic property of the magnetic needle; and were aware that astronomical events occurred in cycles. But the careful investigations of L. A. S´edillot1 have shown that the Chinese made no serious attempt to classify or extend the few rules of arithmetic or geometry with which they were acquainted, or to explain the causes of the phenomena which they observed. The idea that the Chinese had made considerable progress in theoretical mathematics seems to have been due to a misapprehension of the Jesuit missionaries who went to China in the sixteenth century. 1

See Boncompagni’s Bulletino di bibliografia e di storia delle scienze matematiche e fisiche for May, 1868, vol. i, pp. 161–166. On Chinese mathematics, mostly of a later date, see Cantor, chap. xxxi.

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In the first place, they failed to distinguish between the original science of the Chinese and the views which they found prevalent on their arrival—the latter being founded on the work and teaching of Arab or Hindoo missionaries who had come to China in the course of the thirteenth century or later, and while there introduced a knowledge of spherical trigonometry. In the second place, finding that one of the most important government departments was known as the Board of Mathematics, they supposed that its function was to promote and superintend mathematical studies in the empire. Its duties were really confined to the annual preparation of an almanack, the dates and predictions in which regulated many affairs both in public and domestic life. All extant specimens of these almanacks are defective and, in many respects, inaccurate. The only geometrical theorem with which we can be certain that the ancient Chinese were acquainted is that in certain cases (namely, √ when the ratio of the sides is 3 : 4 : 5, or 1 : 1 : 2) the area of the square described on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares described on the sides. It is barely possible that a few geometrical theorems which can be demonstrated in the quasi-experimental way of superposition were also known to them. Their arithmetic was decimal in notation, but their knowledge seems to have been confined to the art of calculation by means of the swan-pan, and the power of expressing the results in writing. Our acquaintance with the early attainments of the Chinese, slight though it is, is more complete than in the case of most of their contemporaries. It is thus specially instructive, and serves to illustrate the fact that a nation may possess considerable skill in the applied arts while they are ignorant of the sciences on which those arts are founded. From the foregoing summary it will be seen that our knowledge of the mathematical attainments of those who preceded the Greeks is very limited; but we may reasonably infer that from one source or another the early Greeks learned the use of the abacus for practical calculations, symbols for recording the results, and as much mathematics as is contained or implied in the Rhind papyrus. It is probable that this sums up their indebtedness to other races. In the next six chapters I shall trace the development of mathematics under Greek influence.

9

FIRST PERIOD.

Mathematics under Greek Influence. This period begins with the teaching of Thales, circ. 600 b.c., and ends with the capture of Alexandria by the Mohammedans in or about 641 a.d. The characteristic feature of this period is the development of Geometry. It will be remembered that I commenced the last chapter by saying that the history of mathematics might be divided into three periods, namely, that of mathematics under Greek influence, that of the mathematics of the middle ages and of the renaissance, and lastly that of modern mathematics. The next four chapters (chapters ii, iii, iv and v) deal with the history of mathematics under Greek influence: to these it will be convenient to add one (chapter vi) on the Byzantine school, since through it the results of Greek mathematics were transmitted to western Europe; and another (chapter vii) on the systems of numeration which were ultimately displaced by the system introduced by the Arabs. I should add that many of the dates mentioned in these chapters are not known with certainty, and must be regarded as only approximately correct. There appeared in December 1921, just before this reprint was struck off, Sir T. L. Heath’s work in 2 volumes on the History of Greek Mathematics. This may now be taken as the standard authority for this period.

10

CHAPTER II. the ionian and pythagorean schools.1 circ. 600 b.c.–400 b.c. With the foundation of the Ionian and Pythagorean schools we emerge from the region of antiquarian research and conjecture into the light of history. The materials at our disposal for estimating the knowledge of the philosophers of these schools previous to about the year 430 b.c. are, however, very scanty Not only have all but fragments of the different mathematical treatises then written been lost, but we possess no copy of the history of mathematics written about 325 b.c. by Eudemus (who was a pupil of Aristotle). Luckily Proclus, who about 450 a.d. wrote a commentary on the earlier part of Euclid’s Elements, was familiar with Eudemus’s work, and freely utilised it in his historical references. We have also a fragment of the General View of Mathematics written by Geminus about 50 b.c., in which the methods of proof used by the early Greek geometricians are compared with those current at a later date. In addition to these general statements we have biographies of a few of the leading mathematicians, and some scattered notes in various writers in which allusions are made to the lives and works of others. The original authorities are criticised and discussed at length in the works mentioned in the footnote to the heading of the chapter. 1

The history of these schools has been discussed by G. Loria in his Le Scienze Esatte nell’ Antica Grecia, Modena, 1893–1900; by Cantor, chaps. v–viii; by G. J. Allman in his Greek Geometry from Thales to Euclid, Dublin, 1889; by J. Gow, in his Greek Mathematics, Cambridge, 1884; by C. A. Bretschneider in his Die Geometrie und die Geometer vor Eukleides, Leipzig, 1870; and partially by H. Hankel in his posthumous Geschichte der Mathematik, Leipzig, 1874.

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The Ionian School. Thales.1 The founder of the earliest Greek school of mathematics and philosophy was Thales, one of the seven sages of Greece, who was born about 640 b.c. at Miletus, and died in the same town about 550 b.c. The materials for an account of his life consist of little more than a few anecdotes which have been handed down by tradition. During the early part of his life Thales was engaged partly in commerce and partly in public affairs; and to judge by two stories that have been preserved, he was then as distinguished for shrewdness in business and readiness in resource as he was subsequently celebrated in science. It is said that once when transporting some salt which was loaded on mules, one of the animals slipping in a stream got its load wet and so caused some of the salt to be dissolved, and finding its burden thus lightened it rolled over at the next ford to which it came; to break it of this trick Thales loaded it with rags and sponges which, by absorbing the water, made the load heavier and soon effectually cured it of its troublesome habit. At another time, according to Aristotle, when there was a prospect of an unusually abundant crop of olives Thales got possession of all the olive-presses of the district; and, having thus “cornered” them, he was able to make his own terms for lending them out, or buying the olives, and thus realized a large sum. These tales may be apocryphal, but it is certain that he must have had considerable reputation as a man of affairs and as a good engineer, since he was employed to construct an embankment so as to divert the river Halys in such a way as to permit of the construction of a ford. Probably it was as a merchant that Thales first went to Egypt, but during his leisure there he studied astronomy and geometry. He was middle-aged when he returned to Miletus; he seems then to have abandoned business and public life, and to have devoted himself to the study of philosophy and science—subjects which in the Ionian, Pythagorean, and perhaps also the Athenian schools, were closely connected: his views on philosophy do not here concern us. He continued to live at Miletus till his death circ. 550 b.c. We cannot form any exact idea as to how Thales presented his geometrical teaching. We infer, however, from Proclus that it consisted of a number of isolated propositions which were not arranged in a logical sequence, but that the proofs were deductive, so that the theorems were 1

See Loria, book I, chap. ii; Cantor, chap. v; Allman, chap. i.

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not a mere statement of an induction from a large number of special instances, as probably was the case with the Egyptian geometricians. The deductive character which he thus gave to the science is his chief claim to distinction. The following comprise the chief propositions that can now with reasonable probability be attributed to him; they are concerned with the geometry of angles and straight lines. (i) The angles at the base of an isosceles triangle are equal (Euc. i, 5). Proclus seems to imply that this was proved by taking another exactly equal isosceles triangle, turning it over, and then superposing it on the first—a sort of experimental demonstration. (ii) If two straight lines cut one another, the vertically opposite angles are equal (Euc. i, 15). Thales may have regarded this as obvious, for Proclus adds that Euclid was the first to give a strict proof of it. (iii) A triangle is determined if its base and base angles be given (cf. Euc. i, 26). Apparently this was applied to find the distance of a ship at sea—the base being a tower, and the base angles being obtained by observation. (iv) The sides of equiangular triangles are proportionals (Euc. vi, 4, or perhaps rather Euc. vi, 2). This is said to have been used by Thales when in Egypt to find the height of a pyramid. In a dialogue given by Plutarch, the speaker, addressing Thales, says, “Placing your stick at the end of the shadow of the pyramid, you made by the sun’s rays two triangles, and so proved that the [height of the] pyramid was to the [length of the] stick as the shadow of the pyramid to the shadow of the stick.” It would seem that the theorem was unknown to the Egyptians, and we are told that the king Amasis, who was present, was astonished at this application of abstract science. (v) A circle is bisected by any diameter. This may have been enunciated by Thales, but it must have been recognised as an obvious fact from the earliest times. (vi) The angle subtended by a diameter of a circle at any point in the circumference is a right angle (Euc. iii, 31). This appears to have been regarded as the most remarkable of the geometrical achievements of Thales, and it is stated that on inscribing a right-angled triangle in a circle he sacrificed an ox to the immortal gods. It has been conjectured that he may have come to this conclusion by noting that the diagonals of a rectangle are equal and bisect one another, and that therefore a rectangle can be inscribed in a circle. If so, and if he went on to apply proposition (i), he would have discovered that the sum of the angles of a

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right-angled triangle is equal to two right angles, a fact with which it is believed that he was acquainted. It has been remarked that the shape of the tiles used in paving floors may have suggested these results. On the whole it seems unlikely that he knew how to draw a perpendicular from a point to a line; but if he possessed this knowledge, it is possible he was also aware, as suggested by some modern commentators, that the sum of the angles of any triangle is equal to two right angles. As far as equilateral and right-angled triangles are concerned, we know from Eudemus that the first geometers proved the general property separately for three species of triangles, and it is not unlikely that they proved it thus. The area about a point can be filled by the angles of six equilateral triangles or tiles, hence the proposition is true for an equilateral triangle. Again, any two equal right-angled triangles can be placed in juxtaposition so as to form a rectangle, the sum of whose angles is four right angles; hence the proposition is true for a right-angled triangle. Lastly, any triangle can be split into the sum of two right-angled triangles by drawing a perpendicular from the biggest angle on the opposite side, and therefore again the proposition is true. The first of these proofs is evidently included in the last, but there is nothing improbable in the suggestion that the early Greek geometers continued to teach the first proposition in the form above given. Thales wrote on astronomy, and among his contemporaries was more famous as an astronomer than as a geometrician. A story runs that one night, when walking out, he was looking so intently at the stars that he tumbled into a ditch, on which an old woman exclaimed, “How can you tell what is going on in the sky when you can’t see what is lying at your own feet?”—an anecdote which was often quoted to illustrate the unpractical character of philosophers. Without going into astronomical details, it may be mentioned that he taught that a year contained about 365 days, and not (as is said to have been previously reckoned) twelve months of thirty days each. It is said that his predecessors occasionally intercalated a month to keep the seasons in their customary places, and if so they must have realized that the year contains, on the average, more than 360 days. There is some reason to think that he believed the earth to be a disc-like body floating on water. He predicted a solar eclipse which took place at or about the time he foretold; the actual date was either May 28, 585 b.c., or September 30, 609 b.c. But though this prophecy and its fulfilment gave extraordinary prestige to his teaching, and secured him the name of one of the seven sages of Greece, it is most likely that he only made

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use of one of the Egyptian or Chaldaean registers which stated that solar eclipses recur at intervals of about 18 years 11 days. Among the pupils of Thales were Anaximander, Anaximenes, Mamercus, and Mandryatus. Of the three mentioned last we know next to nothing. Anaximander was born in 611 b.c., and died in 545 b.c., and succeeded Thales as head of the school at Miletus. According to Suidas he wrote a treatise on geometry in which, tradition says, he paid particular attention to the properties of spheres, and dwelt at length on the philosophical ideas involved in the conception of infinity in space and time. He constructed terrestrial and celestial globes. Anaximander is alleged to have introduced the use of the style or gnomon into Greece. This, in principle, consisted only of a stick stuck upright in a horizontal piece of ground. It was originally used as a sun-dial, in which case it was placed at the centre of three concentric circles, so that every two hours the end of its shadow passed from one circle to another. Such sun-dials have been found at Pompeii and Tusculum. It is said that he employed these styles to determine his meridian (presumably by marking the lines of shadow cast by the style at sunrise and sunset on the same day, and taking the plane bisecting the angle so formed); and thence, by observing the time of year when the noon-altitude of the sun was greatest and least, he got the solstices; thence, by taking half the sum of the noon-altitudes of the sun at the two solstices, he found the inclination of the equator to the horizon (which determined the altitude of the place), and, by taking half their difference, he found the inclination of the ecliptic to the equator. There seems good reason to think that he did actually determine the latitude of Sparta, but it is more doubtful whether he really made the rest of these astronomical deductions. We need not here concern ourselves further with the successors of Thales. The school he established continued to flourish till about 400 b.c., but, as time went on, its members occupied themselves more and more with philosophy and less with mathematics. We know very little of the mathematicians comprised in it, but they would seem to have devoted most of their attention to astronomy. They exercised but slight influence on the further advance of Greek mathematics, which was made almost entirely under the influence of the Pythagoreans, who not only immensely developed the science of geometry, but created a science of numbers. If Thales was the first to direct general attention to geometry, it was Pythagoras, says Proclus, quoting from Eudemus, who

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“changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom and investigated its theorems in an . . . intellectual manner”; and it is accordingly to Pythagoras that we must now direct attention. The Pythagorean School. Pythagoras.1 Pythagoras was born at Samos about 569 b.c., perhaps of Tyrian parents, and died in 500 b.c. He was thus a contemporary of Thales. The details of his life are somewhat doubtful, but the following account is, I think, substantially correct. He studied first under Pherecydes of Syros, and then under Anaximander; by the latter he was recommended to go to Thebes, and there or at Memphis he spent some years. After leaving Egypt he travelled in Asia Minor, and then settled at Samos, where he gave lectures but without much success. About 529 b.c. he migrated to Sicily with his mother, and with a single disciple who seems to have been the sole fruit of his labours at Samos. Thence he went to Tarentum, but very shortly moved to Croton, a Dorian colony in the south of Italy. Here the schools that he opened were crowded with enthusiastic audiences; citizens of all ranks, especially those of the upper classes, attended, and even the women broke a law which forbade their going to public meetings and flocked to hear him. Amongst his most attentive auditors was Theano, the young and beautiful daughter of his host Milo, whom, in spite of the disparity of their ages, he married. She wrote a biography of her husband, but unfortunately it is lost. Pythagoras divided those who attended his lectures into two classes, whom we may term probationers and Pythagoreans. The majority were probationers, but it was only to the Pythagoreans that his chief discoveries were revealed. The latter formed a brotherhood with all things in common, holding the same philosophical and political beliefs, engaged in the same pursuits, and bound by oath not to reveal the teaching or secrets of the school; their food was simple; their discipline 1

See Loria, book I, chap. iii; Cantor, chaps. vi, vii; Allman, chap. ii; Hankel, pp. 92–111; Hoefer, Histoire des math´ematiques, Paris, third edition, 1886, pp. 87– 130; and various papers by S. P. Tannery. For an account of Pythagoras’s life, embodying the Pythagorean traditions, see the biography by Iamblichus, of which there are two or three English translations. Those who are interested in esoteric literature may like to see a modern attempt to reproduce the Pythagorean teaching in Pythagoras, by E. Schur´e, Eng. trans., London, 1906.

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severe; and their mode of life arranged to encourage self-command, temperance, purity, and obedience. This strict discipline and secret organisation gave the society a temporary supremacy in the state which brought on it the hatred of various classes; and, finally, instigated by his political opponents, the mob murdered Pythagoras and many of his followers. Though the political influence of the Pythagoreans was thus destroyed, they seem to have re-established themselves at once as a philosophical and mathematical society, with Tarentum as their headquarters, and they continued to flourish for more than a hundred years. Pythagoras himself did not publish any books; the assumption of his school was that all their knowledge was held in common and veiled from the outside world, and, further, that the glory of any fresh discovery must be referred back to their founder. Thus Hippasus (circ. 470 b.c.) is said to have been drowned for violating his oath by publicly boasting that he had added the dodecahedron to the number of regular solids enumerated by Pythagoras. Gradually, as the society became more scattered, this custom was abandoned, and treatises containing the substance of their teaching and doctrines were written. The first book of the kind was composed, about 370 b.c., by Philolaus, and we are told that Plato secured a copy of it. We may say that during the early part of the fifth century before Christ the Pythagoreans were considerably in advance of their contemporaries, but by the end of that time their more prominent discoveries and doctrines had become known to the outside world, and the centre of intellectual activity was transferred to Athens. Though it is impossible to separate precisely the discoveries of Pythagoras himself from those of his school of a later date, we know from Proclus that it was Pythagoras who gave geometry that rigorous character of deduction which it still bears, and made it the foundation of a liberal education; and there is reason to believe that he was the first to arrange the leading propositions of the subject in a logical order. It was also, according to Aristoxenus, the glory of his school that they raised arithmetic above the needs of merchants. It was their boast that they sought knowledge and not wealth, or in the language of one of their maxims, “a figure and a step forwards, not a figure to gain three oboli.” Pythagoras was primarily a moral reformer and philosopher, but his system of morality and philosophy was built on a mathematical foundation. His mathematical researches were, however, designed to lead

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up to a system of philosophy whose exposition was the main object of his teaching. The Pythagoreans began by dividing the mathematical subjects with which they dealt into four divisions: numbers absolute or arithmetic, numbers applied or music, magnitudes at rest or geometry, and magnitudes in motion or astronomy. This “quadrivium” was long considered as constituting the necessary and sufficient course of study for a liberal education. Even in the case of geometry and arithmetic (which are founded on inferences unconsciously made and common to all men) the Pythagorean presentation was involved with philosophy; and there is no doubt that their teaching of the sciences of astronomy, mechanics, and music (which can rest safely only on the results of conscious observation and experiment) was intermingled with metaphysics even more closely. It will be convenient to begin by describing their treatment of geometry and arithmetic. First, as to their geometry. Pythagoras probably knew and taught the substance of what is contained in the first two books of Euclid about parallels, triangles, and parallelograms, and was acquainted with a few other isolated theorems including some elementary propositions on irrational magnitudes; but it is suspected that many of his proofs were not rigorous, and in particular that the converse of a theorem was sometimes assumed without a proof. It is hardly necessary to say that we are unable to reproduce the whole body of Pythagorean teaching on this subject, but we gather from the notes of Proclus on Euclid, and from a few stray remarks in other writers, that it included the following propositions, most of which are on the geometry of areas. (i) It commenced with a number of definitions, which probably were rather statements connecting mathematical ideas with philosophy than explanations of the terms used. One has been preserved in the definition of a point as unity having position. (ii) The sum of the angles of a triangle was shown to be equal to two right angles (Euc. i, 32); and in the proof, which has been preserved, the results of the propositions Euc. i, 13 and the first part of Euc. i, 29 are quoted. The demonstration is substantially the same as that in Euclid, and it is most likely that the proofs there given of the two propositions last mentioned are also due to Pythagoras himself. (iii) Pythagoras certainly proved the properties of right-angled triangles which are given in Euc. i, 47 and i, 48. We know that the proofs of these propositions which are found in Euclid were of Euclid’s own invention; and a good deal of curiosity has been excited to discover what was the demonstration which was originally offered by Pythago-

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ras of the first of these theorems. It has been conjectured that not improbably it may have been one of the two following.1 A

F

E

B

K

G D

H

C

(α) Any square ABCD can be split up, as in Euc. ii, 4, into two squares BK and DK and two equal rectangles AK and CK: that is, it is equal to the square on F K, the square on EK, and four times the triangle AEF . But, if points be taken, G on BC, H on CD, and E on DA, so that BG, CH, and DE are each equal to AF , it can be easily shown that EF GH is a square, and that the triangles AEF , BF G, CGH, and DHE are equal: thus the square ABCD is also equal to the square on EF and four times the triangle AEF . Hence the square on EF is equal to the sum of the squares on F K and EK. A

B

D

C

(β) Let ABC be a right-angled triangle, A being the right angle. Draw AD perpendicular to BC. The triangles ABC and DBA are 1

A collection of a hundred proofs of Euc. i, 47 was published in the American Mathematical Monthly Journal, vols. iii. iv. v. vi. 1896–1899.

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similar,

Similarly Hence

∴ BC : AB = AB : BD. BC : AC = AC : DC. 2 AB + AC 2 = BC(BD + DC) = BC 2 .

This proof requires a knowledge of the results of Euc. ii, 2, vi, 4, and vi, 17, with all of which Pythagoras was acquainted. (iv) Pythagoras is credited by some writers with the discovery of the theorems Euc. i, 44, and i, 45, and with giving a solution of the problem Euc. ii, 14. It is said that on the discovery of the necessary construction for the problem last mentioned he sacrificed an ox, but as his school had all things in common the liberality was less striking than it seems at first. The Pythagoreans of a later date were aware of the extension given in Euc. vi, 25, and Allman thinks that Pythagoras himself was acquainted with it, but this must be regarded as doubtful. It will be noticed that Euc. ii, 14 provides a geometrical solution of the equation x2 = ab. (v) Pythagoras showed that the plane about a point could be completely filled by equilateral triangles, by squares, or by regular hexagons —results that must have been familiar wherever tiles of these shapes were in common use. (vi) The Pythagoreans were said to have attempted the quadrature of the circle: they stated that the circle was the most perfect of all plane figures. (vii) They knew that there were five regular solids inscribable in a sphere, which was itself, they said, the most perfect of all solids. (viii) From their phraseology in the science of numbers and from other occasional remarks, it would seem that they were acquainted with the methods used in the second and fifth books of Euclid, and knew something of irrational magnitudes. In particular, there is reason to believe that Pythagoras proved that the side and the diagonal of a square were incommensurable, and that it was this discovery which led the early Greeks to banish the conceptions of number and measurement from their geometry. A proof of this proposition which may be that due to Pythagoras is given below.1 1

See below, page 49.

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Next, as to their theory of numbers.1 In this Pythagoras was chiefly concerned with four different groups of problems which dealt respectively with polygonal numbers, with ratio and proportion, with the factors of numbers, and with numbers in series; but many of his arithmetical inquiries, and in particular the questions on polygonal numbers and proportion, were treated by geometrical methods. H

K

A

C

L

Pythagoras commenced his theory of arithmetic by dividing all numbers into even or odd: the odd numbers being termed gnomons. An odd number, such as 2n + 1, was regarded as the difference of two square numbers (n + 1)2 and n2 ; and the sum of the gnomons from 1 to 2n + 1 was stated to be a square number, viz. (n + 1)2 , its square root was termed a side. Products of two numbers were called plane, and if a product had no exact square root it was termed an oblong. A product of three numbers was called a solid number, and, if the three numbers were equal, a cube. All this has obvious reference to geometry, and the opinion is confirmed by Aristotle’s remark that when a gnomon is put round a square the figure remains a square though it is increased in dimensions. Thus, in the figure given above in which n is taken equal to 5, the gnomon AKC (containing 11 small squares) when put round the square AC (containing 52 small squares) makes a square HL (containing 62 small squares). It is possible that several 1

See the appendix Sur l’arithm´etique pythagorienne to S. P. Tannery’s La science hell`ene, Paris, 1887.

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of the numerical theorems due to Greek writers were discovered and proved by an analogous method: the abacus can be used for many of these demonstrations. The numbers (2n2 + 2n + 1), (2n2 + 2n), and (2n + 1) possessed special importance as representing the hypotenuse and two sides of a right-angled triangle: Cantor thinks that Pythagoras knew this fact before discovering the geometrical proposition Euc. i, 47. A more general expression for such numbers is (m2 +n2 ), 2mn, and (m2 −n2 ), or multiples of them: it will be noticed that the result obtained by Pythagoras can be deduced from these expressions by assuming m = n + 1; at a later time Archytas and Plato gave rules which are equivalent to taking n = 1; Diophantus knew the general expressions. After this preliminary discussion the Pythagoreans proceeded to the four special problems already alluded to. Pythagoras was himself acquainted with triangular numbers; polygonal numbers of a higher order were discussed by later members of the school. A triangular number represents the sum of a number of counters laid in rows on a plane; the bottom row containing n, and each succeeding row one less: it is therefore equal to the sum of the series n + (n − 1) + (n − 2) + . . . + 2 + 1, that is, to 21 n(n + 1). Thus the triangular number corresponding to 4 is 10. This is the explanation of the language of Pythagoras in the wellknown passage in Lucian where the merchant asks Pythagoras what he can teach him. Pythagoras replies “I will teach you how to count.” Merchant, “I know that already.” Pythagoras, “How do you count?” Merchant, “One, two, three, four—” Pythagoras, “Stop! what you take to be four is ten, a perfect triangle and our symbol.” The Pythagoreans are, on somewhat doubtful authority, said to have classified numbers by comparing them with the sum of their integral subdivisors or factors, calling a number excessive, perfect, or defective, according as the sum of these subdivisors was greater than, equal to, or less than the number: the classification at first being restricted to even numbers. The third group of problems which they considered dealt with numbers which formed a proportion; presumably these were discussed with the aid of geometry as is done in the fifth book of Euclid. Lastly, the Pythagoreans were concerned with series of numbers in arithmetical, geometrical, harmonical, and musical progressions. The three progressions first mentioned are well known; four integers are said to be in musical pro-

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gression when they are in the ratio a : 2ab/(a + b) : 12 (a + b) : b, for example, 6, 8, 9, and 12 are in musical progression. Of the Pythagorean treatment of the applied subjects of the quadrivium, and the philosophical theories founded on them, we know very little. It would seem that Pythagoras was much impressed by certain numerical relations which occur in nature. It has been suggested that he was acquainted with some of the simpler facts of crystallography. It is thought that he was aware that the notes sounded by a vibrating string depend on the length of the string, and in particular that lengths which gave a note, its fifth and its octave were in the ratio 2 : 3 : 4, forming terms in a musical progression. It would seem, too, that he believed that the distances of the astrological planets from the earth were also in musical progression, and that the heavenly bodies in their motion through space gave out harmonious sounds: hence the phrase the harmony of the spheres. These and similar conclusions seem to have suggested to him that the explanation of the order and harmony of the universe was to be found in the science of numbers, and that numbers are to some extent the cause of form as well as essential to its accurate measurement. He accordingly proceeded to attribute particular properties to particular numbers and geometrical figures. For example, he taught that the cause of colour was to be sought in properties of the number five, that the explanation of fire was to be discovered in the nature of the pyramid, and so on. I should not have alluded to this were it not that the Pythagorean tradition strengthened, or perhaps was chiefly responsible for the tendency of Greek writers to found the study of nature on philosophical conjectures and not on experimental observation—a tendency to which the defects of Hellenic science must be largely attributed. After the death of Pythagoras his teaching seems to have been carried on by Epicharmus and Hippasus, and subsequently by Philolaus (specially distinguished as an astronomer), Archippus, and Lysis. About a century after the murder of Pythagoras we find Archytas recognised as the head of the school. Archytas.1 Archytas, circ. 400 b.c., was one of the most influential citizens of Tarentum, and was made governor of the city no less 1

See Allman, chap. iv. A catalogue of the works of Archytas is given by Fabricius in his Bibliotheca Graeca, vol. i, p. 833: most of the fragments on philosophy were published by Thomas Gale in his Opuscula Mythologica, Cambridge, 1670; and by Thomas Taylor as an Appendix to his translation of Iamblichus’s Life of Pythagoras, London, 1818. See also the references given by Cantor, vol. i, p. 203.

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than seven times. His influence among his contemporaries was very great, and he used it with Dionysius on one occasion to save the life of Plato. He was noted for the attention he paid to the comfort and education of his slaves and of children in the city. He was drowned in a shipwreck near Tarentum, and his body washed on shore—a fit punishment, in the eyes of the more rigid Pythagoreans, for his having departed from the lines of study laid down by their founder. Several of the leaders of the Athenian school were among his pupils and friends, and it is believed that much of their work was due to his inspiration. The Pythagoreans at first made no attempt to apply their knowledge to mechanics, but Archytas is said to have treated it with the aid of geometry. He is alleged to have invented and worked out the theory of the pulley, and is credited with the construction of a flying bird and some other ingenious mechanical toys. He introduced various mechanical devices for constructing curves and solving problems. These were objected to by Plato, who thought that they destroyed the value of geometry as an intellectual exercise, and later Greek geometricians confined themselves to the use of two species of instruments, namely, rulers and compasses. Archytas was also interested in astronomy; he taught that the earth was a sphere rotating round its axis in twenty-four hours, and round which the heavenly bodies moved. Archytas was one of the first to give a solution of the problem to duplicate a cube, that is, to find the side of a cube whose volume is double that of a given cube. This was one of the most famous problems of antiquity.1 The construction given by Archytas is equivalent to the following. On the diameter OA of the base of a right circular cylinder describe a semicircle whose plane is perpendicular to the base of the cylinder. Let the plane containing this semicircle rotate round the generator through O, then the surface traced out by the semicircle will cut the cylinder in a tortuous curve. This curve will be cut by a right cone whose axis is OA and semivertical angle is (say) 60◦ in a point P , such that the projection of OP on the base of the cylinder will be to the radius of the cylinder in the ratio of the side of the required cube to that of the given cube. The proof given by Archytas is of course geometrical;2 it will be enough here to remark that in the course of it he shews himself acquainted with the results of the propositions Euc. iii, 18, Euc. iii, 35, and Euc. xi, 19. To shew analytically that the construction is correct, 1 2

See below, pp. 30, 34, 34. It is printed by Allman, pp. 111–113.

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take OA as the axis of x, and the generator through O as axis of z, then, with the usual notation in polar co-ordinates, and if a be the radius of the cylinder, we have for the equation of the surface described by the semicircle, r = 2a sin θ; for that of the cylinder, r sin θ = 2a cos φ; and for that of the cone, sin θ cos φ = 12 . These three surfaces cut in a point such that sin3 θ = 12 , and, therefore, if ρ be the projection of OP on the base of the cylinder, then ρ3 = (r sin θ)3 = 2a3 . Hence the volume of the cube whose side is ρ is twice that of a cube whose side is a. I mention the problem and give the construction used by Archytas to illustrate how considerable was the knowledge of the Pythagorean school at the time. Theodorus. Another Pythagorean of about the same date as Archytas was Theodorus of Cyrene, who is√said√to have √ proved √ √geomet√ 3, 5, 6, 7, 8, 10, rically that the numbers represented by √ √ √ √ √ √ 11, 12, 13, 14, 15, and 17 are incommensurable with unity. Theaetetus was one of his pupils. Perhaps Timaeus of Locri and Bryso of Heraclea should be mentioned as other distinguished Pythagoreans of this time. It is believed that Bryso attempted to find the area of a circle by inscribing and circumscribing squares, and finally obtained polygons between whose areas the area of the circle lay; but it is said that at some point he assumed that the area of the circle was the arithmetic mean between an inscribed and a circumscribed polygon. Other Greek Mathematical Schools in the Fifth Century b.c. It would be a mistake to suppose that Miletus and Tarentum were the only places where, in the fifth century, Greeks were engaged in laying a scientific foundation for the study of mathematics. These towns represented the centres of chief activity, but there were few cities or colonies of any importance where lectures on philosophy and geometry were not given. Among these smaller schools I may mention those at Chios, Elea, and Thrace. The best known philosopher of the School of Chios was Oenopides, who was born about 500 b.c., and died about 430 b.c. He devoted himself chiefly to astronomy, but he had studied geometry in Egypt, and is credited with the solution of two problems, namely, to draw a straight line from a given external point perpendicular to a given straight line (Euc. i, 12), and at a given point to construct an angle equal to a given angle (Euc. i, 23).

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Another important centre was at Elea in Italy. This was founded in Sicily by Xenophanes. He was followed by Parmenides, Zeno, and Melissus. The members of the Eleatic School were famous for the difficulties they raised in connection with questions that required the use of infinite series, such, for example, as the well-known paradox of Achilles and the tortoise, enunciated by Zeno, one of their most prominent members. Zeno was born in 495 b.c., and was executed at Elea in 435 b.c. in consequence of some conspiracy against the state; he was a pupil of Parmenides, with whom he visited Athens, circ. 455– 450 b.c. Zeno argued that if Achilles ran ten times as fast as a tortoise, yet if the tortoise had (say) 1000 yards start it could never be overtaken: for, when Achilles had gone the 1000 yards, the tortoise would still be 100 yards in front of him; by the time he had covered these 100 yards, it would still be 10 yards in front of him; and so on for ever: thus Achilles would get nearer and nearer to the tortoise, but never overtake it. The fallacy is usually explained by the argument that the time required to overtake the tortoise, can be divided into an infinite number of parts, as stated in the question, but these get smaller and smaller in geometrical progression, and the sum of them all is a finite time: after the lapse of that time Achilles would be in front of the tortoise. Probably Zeno would have replied that this argument rests on the assumption that space is infinitely divisible, which is the question under discussion: he himself asserted that magnitudes are not infinitely divisible. These paradoxes made the Greeks look with suspicion on the use of infinitesimals, and ultimately led to the invention of the method of exhaustions. The Atomistic School, having its headquarters in Thrace, was another important centre. This was founded by Leucippus, who was a pupil of Zeno. He was succeeded by Democritus and Epicurus. Its most famous mathematician was Democritus, born at Abdera in 460 b.c., and said to have died in 370 b.c., who, besides philosophical works, wrote on plane and solid geometry, incommensurable lines, perspective, and numbers. These works are all lost. From the Archimedean MS., discovered by Heiberg in 1906, it would seem that Democritus enunciated, but without a proof, the proposition that the volume of a pyramid is equal to one-third that of a prism of an equal base and of equal height. But though several distinguished individual philosophers may be mentioned who, during the fifth century, lectured at different cities,

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they mostly seem to have drawn their inspiration from Tarentum, and towards the end of the century to have looked to Athens as the intellectual capital of the Greek world; and it is to the Athenian schools that we owe the next great advance in mathematics.

27

CHAPTER III. the schools of athens and cyzicus.1 circ. 420 b.c.–300 b.c. It was towards the close of the fifth century before Christ that Athens first became the chief centre of mathematical studies. Several causes conspired to bring this about. During that century she had become, partly by commerce, partly by appropriating for her own purposes the contributions of her allies, the most wealthy city in Greece; and the genius of her statesmen had made her the centre on which the politics of the peninsula turned. Moreover, whatever states disputed her claim to political supremacy her intellectual pre-eminence was admitted by all. There was no school of thought which had not at some time in that century been represented at Athens by one or more of its leading thinkers; and the ideas of the new science, which was being so eagerly studied in Asia Minor and Graecia Magna, had been brought before the Athenians on various occasions. Anaxagoras. Amongst the most important of the philosophers who resided at Athens and prepared the way for the Athenian school I may mention Anaxagoras of Clazomenae, who was almost the last philosopher of the Ionian school. He was born in 500 b.c., and died in 428 b.c. He seems to have settled at Athens about 440 b.c., and there taught the results of the Ionian philosophy. Like all members of that school he was much interested in astronomy. He asserted that 1

The history of these schools is discussed at length in G. Loria’s Le Scienze Esatte nell’ Antica Grecia, Modena, 1893–1900; in G. J. Allman’s Greek Geometry from Thales to Euclid, Dublin, 1889; and in J. Gow’s Greek Mathematics, Cambridge, 1884; it is also treated by Cantor, chaps. ix, x, and xi; by Hankel, pp. 111–156; and by C. A. Bretschneider in his Die Geometrie und die Geometer vor Eukleides, Leipzig, 1870; a critical account of the original authorities is given by S. P. Tannery in his G´eom´etrie Grecque, Paris, 1887, and other papers.

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the sun was larger than the Peloponnesus: this opinion, together with some attempts he had made to explain various physical phenomena which had been previously supposed to be due to the direct action of the gods, led to a prosecution for impiety, and he was convicted. While in prison he is said to have written a treatise on the quadrature of the circle. The Sophists. The sophists can hardly be considered as belonging to the Athenian school, any more than Anaxagoras can; but like him they immediately preceded and prepared the way for it, so that it is desirable to devote a few words to them. One condition for success in public life at Athens was the power of speaking well, and as the wealth and power of the city increased a considerable number of “sophists” settled there who undertook amongst other things to teach the art of oratory. Many of them also directed the general education of their pupils, of which geometry usually formed a part. We are told that two of those who are usually termed sophists made a special study of geometry—these were Hippias of Elis and Antipho, and one made a special study of astronomy—this was Meton, after whom the metonic cycle is named. Hippias. The first of these geometricians, Hippias of Elis (circ. 420 b.c.), is described as an expert arithmetician, but he is best known to us through his invention of a curve called the quadratrix, by means of which an angle can be trisected, or indeed divided in any given ratio. If the radius of a circle rotate uniformly round the centre O from the position OA through a right angle to OB, and in the same time a straight line drawn perpendicular to OB move uniformly parallel to itself from the position OA to BC, the locus of their intersection will be the quadratrix. Let OR and M Q be the position of these lines at any time; and let them cut in P , a point on the curve. Then angle AOP : angle AOB = OM : OB. Similarly, if OR0 be another position of the radius, angle AOP 0 : angle AOB = OM 0 : OB ∴ angle AOP : angle AOP 0 = OM : OM 0 ; ∴ angle AOP 0 : angle P 0 OP = OM 0 : M M. Hence, if the angle AOP be given, and it be required to divide it in any given ratio, it is sufficient to divide OM in that ratio at M 0 and draw the line M 0 P 0 ; then OP 0 will divide AOP in the required ratio.

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B

29

C R

M

M0

Q

P

R0 P0

O

A

If OA be taken as the initial line, OP = r, the angle AOP = θ, and OA = a, we have θ : 12 π = r sin θ : a, and the equation of the curve is πr = 2aθ cosec θ. Hippias devised an instrument to construct the curve mechanically; but constructions which involved the use of any mathematical instruments except a ruler and a pair of compasses were objected to by Plato, and rejected by most geometricians of a subsequent date. Antipho. The second sophist whom I mentioned was Antipho (circ. 420 b.c.). He is one of the very few writers among the ancients who attempted to find the area of a circle by considering it as the limit of an inscribed regular polygon with an infinite number of sides. He began by inscribing an equilateral triangle (or, according to some accounts, a square); on each side he inscribed in the smaller segment an isosceles triangle, and so on ad infinitum. This method of attacking the quadrature problem is similar to that described above as used by Bryso of Heraclea. No doubt there were other cities in Greece besides Athens where similar and equally meritorious work was being done, though the record of it has now been lost; I have mentioned here the investigations of these three writers, chiefly because they were the immediate predecessors of those who created the Athenian school. The Schools of Athens and Cyzicus. The history of the Athenian school begins with the teaching of Hippocrates about 420 b.c.; the school was established on a permanent basis by the labours

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of Plato and Eudoxus; and, together with the neighbouring school of Cyzicus, continued to extend on the lines laid down by these three geometricians until the foundation (about 300 b.c.) of the university at Alexandria drew thither most of the talent of Greece. Eudoxus, who was amongst the most distinguished of the Athenian mathematicians, is also reckoned as the founder of the school at Cyzicus. The connection between this school and that of Athens was very close, and it is now impossible to disentangle their histories. It is said that Hippocrates, Plato, and Theaetetus belonged to the Athenian school; while Eudoxus, Menaechmus, and Aristaeus belonged to that of Cyzicus. There was always a constant intercourse between the two schools, the earliest members of both had been under the influence either of Archytas or of his pupil Theodorus of Cyrene, and there was no difference in their treatment of the subject, so that they may be conveniently treated together. Before discussing the work of the geometricians of these schools in detail I may note that they were especially interested in three problems:1 namely (i), the duplication of a cube, that is, the determination of the side of a cube whose volume is double that of a given cube; (ii) the trisection of an angle; and (iii) the squaring of a circle, that is, the determination of a square whose area is equal to that of a given circle. Now the first two of these problems (considered analytically) require the solution of a cubic equation; and, since a construction by means of circles (whose equations are of the form x2 + y 2 + ax + by + c = 0) and straight lines (whose equations are of the form x + βy + γ = 0) cannot be equivalent to the solution of a cubic equation, the problems are insoluble if in our constructions we restrict ourselves to the use of circles and straight lines, that is, to Euclidean geometry. If the use of the conic sections be permitted, both of these questions can be solved in many ways. The third problem is equivalent to finding a rectangle whose sides are equal respectively to the radius and to the semiperimeter of the circle. These lines have been long known to be incommensurable, but it is only recently that it has been shewn by Lindemann that their ratio cannot be the root of a rational algebraical equation. Hence this problem also is insoluble by Euclidean geometry. The Athenians and Cyzicians were thus destined to fail in all three 1

On these problems, solutions of them, and the authorities for their history, see my Mathematical Recreations and Problems, London, ninth edition, 1920, chap. xiv.

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problems, but the attempts to solve them led to the discovery of many new theorems and processes. Besides attacking these problems the later Platonic school collected all the geometrical theorems then known and arranged them systematically. These collections comprised the bulk of the propositions in Euclid’s Elements, books i–ix, xi, and xii, together with some of the more elementary theorems in conic sections. Hippocrates. Hippocrates of Chios (who must be carefully distinguished from his contemporary, Hippocrates of Cos, the celebrated physician) was one of the greatest of the Greek geometricians. He was born about 470 b.c. at Chios, and began life as a merchant. The accounts differ as to whether he was swindled by the Athenian customhouse officials who were stationed at the Chersonese, or whether one of his vessels was captured by an Athenian pirate near Byzantium; but at any rate somewhere about 430 b.c. he came to Athens to try to recover his property in the law courts. A foreigner was not likely to succeed in such a case, and the Athenians seem only to have laughed at him for his simplicity, first in allowing himself to be cheated, and then in hoping to recover his money. While prosecuting his cause he attended the lectures of various philosophers, and finally (in all probability to earn a livelihood) opened a school of geometry himself. He seems to have been well acquainted with the Pythagorean philosophy, though there is no sufficient authority for the statement that he was ever initiated as a Pythagorean. He wrote the first elementary text-book of geometry, a text-book on which probably Euclid’s Elements was founded; and therefore he may be said to have sketched out the lines on which geometry is still taught in English schools. It is supposed that the use of letters in diagrams to describe a figure was made by him or introduced about this time, as he employs expressions such as “the point on which the letter A stands” and “the line on which AB is marked.” Cantor, however, thinks that the Pythagoreans had previously been accustomed to represent the five vertices of the pentagram-star by the letters υ γ ι θ α; and though this was a single instance, perhaps they may have used the method generally. The Indian geometers never employed letters to aid them in the description of their figures. Hippocrates also denoted the square on a line by the word δύναμις, and thus gave the technical meaning to the word power which it still retains in algebra: there is reason to think that this use of the word was derived from the Pythagoreans, who are said to have enunciated the result of the proposition Euc. i, 47, in the

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form that “the total power of the sides of a right-angled triangle is the same as that of the hypotenuse.” In this text-book Hippocrates introduced the method of “reducing” one theorem to another, which being proved, the thing proposed necessarily follows; of this method the reductio ad absurdum is an illustration. No doubt the principle had been used occasionally before, but he drew attention to it as a legitimate mode of proof which was capable of numerous applications. He elaborated the geometry of the circle: proving, among other propositions, that similar segments of a circle contain equal angles; that the angle subtended by the chord of a circle is greater than, equal to, or less than a right angle as the segment of the circle containing it is less than, equal to, or greater than a semicircle (Euc. iii, 31); and probably several other of the propositions in the third book of Euclid. It is most likely that he also established the propositions that [similar] circles are to one another as the squares of their diameters (Euc. xii, 2), and that similar segments are as the squares of their chords. The proof given in Euclid of the first of these theorems is believed to be due to Hippocrates. The most celebrated discoveries of Hippocrates were, however, in connection with the quadrature of the circle and the duplication of the cube, and owing to his influence these problems played a prominent part in the history of the Athenian school. The following propositions will sufficiently illustrate the method by which he attacked the quadrature problem. A

F G

B

D E

O

C

(α) He commenced by finding the area of a lune contained between a semicircle and a quadrantal arc standing on the same chord. This he did as follows. Let ABC be an isosceles right-angled triangle inscribed in the semicircle ABOC, whose centre is O. On AB and AC as diameters

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describe semicircles as in the figure. Then, since by Euc. i, 47, sq. on BC = sq. on AC + sq. on AB, therefore, by Euc. xii, 2, area

1 2

on BC = area

1 2

on AC + area

1 2

on AB.

Take away the common parts ∴ area 4ABC = sum of areas of lunes AECD and AF BG. Hence the area of the lune AECD is equal to half that of the triangle ABC. B

C

E F

D

O

A

(β) He next inscribed half a regular hexagon ABCD in a semicircle whose centre was O, and on OA, AB, BC, and CD as diameters described semicircles of which those on OA and AB are drawn in the figure. Then AD is double any of the lines OA, AB, BC, and CD, ∴ sq. on AD = sum of sqs. on OA, AB, BC, and CD, ∴ area ABCD = sum of areas of 12 s on OA, AB, BC, and CD. 1 2

Take away the common parts ∴ area trapezium ABCD = 3 lune AEBF + 21 on OA.

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If therefore the area of this latter lune be known, so is that of the semicircle described on OA as diameter. According to Simplicius, Hippocrates assumed that the area of this lune was the same as the area of the lune found in proposition (α); if this be so, he was of course mistaken, as in this case he is dealing with a lune contained between a semicircle and a sextantal arc standing on the same chord; but it seems more probable that Simplicius misunderstood Hippocrates. Hippocrates also enunciated various other theorems connected with lunes (which have been collected by Bretschneider and by Allman) of which the theorem last given is a typical example. I believe that they are the earliest instances in which areas bounded by curves were determined by geometry. The other problem to which Hippocrates turned his attention was the duplication of a cube, that is, the determination of the side of a cube whose volume is double that of a given cube. This problem was known in ancient times as the Delian problem, in consequence of a legend that the Delians had consulted Plato on the subject. In one form of the story, which is related by Philoponus, it is asserted that the Athenians in 430 b.c., when suffering from the plague of eruptive typhoid fever, consulted the oracle at Delos as to how they could stop it. Apollo replied that they must double the size of his altar which was in the form of a cube. To the unlearned suppliants nothing seemed more easy, and a new altar was constructed either having each of its edges double that of the old one (from which it followed that the volume was increased eightfold) or by placing a similar cubic altar next to the old one. Whereupon, according to the legend, the indignant god made the pestilence worse than before, and informed a fresh deputation that it was useless to trifle with him, as his new altar must be a cube and have a volume exactly double that of his old one. Suspecting a mystery the Athenians applied to Plato, who referred them to the geometricians, and especially to Euclid, who had made a special study of the problem. The introduction of the names of Plato and Euclid is an obvious anachronism. Eratosthenes gives a somewhat similar account of its origin, but with king Minos as the propounder of the problem. Hippocrates reduced the problem of duplicating the cube to that of finding two means between one straight line (a), and another twice as long (2a). If these means be x and y, we have a : x = x : y = y : 2a, from which it follows that x3 = 2a3 . It is in this form that the problem is usually presented now. Hippocrates did not succeed in finding a construction for these means.

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Plato. The next philosopher of the Athenian school who requires mention here was Plato. He was born at Athens in 429 b.c., and was, as is well known, a pupil for eight years of Socrates; much of the teaching of the latter is inferred from Plato’s dialogues. After the execution of his master in 399 b.c. Plato left Athens, and being possessed of considerable wealth he spent some years in travelling; it was during this time that he studied mathematics. He visited Egypt with Eudoxus, and Strabo says that in his time the apartments they occupied at Heliopolis were still shewn. Thence Plato went to Cyrene, where he studied under Theodorus. Next he moved to Italy, where he became intimate with Archytas the then head of the Pythagorean school, Eurytas of Metapontum, and Timaeus of Locri. He returned to Athens about the year 380 b.c., and formed a school of students in a suburban gymnasium called the “Academy.” He died in 348 b.c. Plato, like Pythagoras, was primarily a philosopher, and perhaps his philosophy should be regarded as founded on the Pythagorean rather than on the Socratic teaching. At any rate it, like that of the Pythagoreans, was coloured with the idea that the secret of the universe is to be found in number and in form; hence, as Eudemus says, “he exhibited on every occasion the remarkable connection between mathematics and philosophy.” All the authorities agree that, unlike many later philosophers, he made a study of geometry or some exact science an indispensable preliminary to that of philosophy. The inscription over the entrance to his school ran “Let none ignorant of geometry enter my door,” and on one occasion an applicant who knew no geometry is said to have been refused admission as a student. Plato’s position as one of the masters of the Athenian mathematical school rests not so much on his individual discoveries and writings as on the extraordinary influence he exerted on his contemporaries and successors. Thus the objection that he expressed to the use in the construction of curves of any instruments other than rulers and compasses was at once accepted as a canon which must be observed in such problems. It is probably due to Plato that subsequent geometricians began the subject with a carefully compiled series of definitions, postulates, and axioms. He also systematized the methods which could be used in attacking mathematical questions, and in particular directed attention to the value of analysis. The analytical method of proof begins by assuming that the theorem or problem is solved, and thence deducing some result: if the result be false, the theorem is not true or the problem is incapable of solution: if the result be true, and if the

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steps be reversible, we get (by reversing them) a synthetic proof; but if the steps be not reversible, no conclusion can be drawn. Numerous illustrations of the method will be found in any modern text-book on geometry. If the classification of the methods of legitimate induction given by Mill in his work on logic had been universally accepted and every new discovery in science had been justified by a reference to the rules there laid down, he would, I imagine, have occupied a position in reference to modern science somewhat analogous to that which Plato occupied in regard to the mathematics of his time. The following is the only extant theorem traditionally attributed to Plato. If CAB and DAB be two right-angled triangles, having one side, AB, common, their other sides, AD and BC, parallel, and their hypotenuses, AC and BD, at right angles, then, if these hypotenuses cut in P , we have P C : P B = P B : P A = P A : P D. This theorem was used in duplicating the cube, for, if such triangles can be constructed having P D = 2P C, the problem will be solved. It is easy to make an instrument by which the triangles can be constructed. Eudoxus.1 Of Eudoxus, the third great mathematician of the Athenian school and the founder of that at Cyzicus, we know very little. He was born in Cnidus in 408 b.c. Like Plato, he went to Tarentum and studied under Archytas the then head of the Pythagoreans. Subsequently he travelled with Plato to Egypt, and then settled at Cyzicus, where he founded the school of that name. Finally he and his pupils moved to Athens. There he seems to have taken some part in public affairs, and to have practised medicine; but the hostility of Plato and his own unpopularity as a foreigner made his position uncomfortable, and he returned to Cyzicus or Cnidus shortly before his death. He died while on a journey to Egypt in 355 b.c. His mathematical work seems to have been of a high order of excellence. He discovered most of what we now know as the fifth book of Euclid, and proved it in much the same form as that in which it is there given. He discovered some theorems on what was called “the golden section.” A H B The problem to cut a line AB in the golden section, that is, to divide it, say at H, in extreme and mean ratio 1

The works of Eudoxus were discussed in considerable detail by H. K¨ unssberg of Dinkelsb¨ uhl in 1888 and 1890; see also the authorities mentioned above in the footnote on p. 27.

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(that is, so that AB : AH = AH : HB) is solved in Euc. ii, 11, and probably was known to the Pythagoreans at an early date. If we denote AB by l, AH by a, and HB by b, the theorems that Eudoxus proved are equivalent to the following algebraical identities. (i) (a + 21 l)2 = 5( 12 l)2 . (ii) Conversely, if (i) be true, and AH be taken equal to a, then AB will be divided at H in a golden section. (iii) (b + 21 a)2 = 5( 12 a2 ). (iv) l2 + b2 = 3a2 . (v) l + a : l = l : a, which gives another golden section. These propositions were subsequently put by Euclid as the first five propositions of his thirteenth book, but they might have been equally well placed towards the end of the second book. All of them are obvious algebraically, since l = a + b and a2 = bl. Eudoxus further established the “method of exhaustions”; which depends on the proposition that “if from the greater of two unequal magnitudes there be taken more than its half, and from the remainder more than its half, and so on, there will at length remain a magnitude less than the least of the proposed magnitudes.” This proposition was placed by Euclid as the first proposition of the tenth book of his Elements, but in most modern school editions it is printed at the beginning of the twelfth book. By the aid of this theorem the ancient geometers were able to avoid the use of infinitesimals: the method is rigorous, but awkward of application. A good illustration of its use is to be found in the demonstration of Euc. xii, 2, namely, that the square of the radius of one circle is to the square of the radius of another circle as the area of the first circle is to an area which is neither less nor greater than the area of the second circle, and which therefore must be exactly equal to it: the proof given by Euclid is (as was usual) completed by a reductio ad absurdum. Eudoxus applied the principle to shew that the volume of a pyramid or a cone is one-third that of the prism or the cylinder on the same base and of the same altitude (Euc. xii, 7 and 10). It is believed that he proved that the volumes of two spheres were to one another as the cubes of their radii; some writers attribute the proposition Euc. xii, 2 to him, and not to Hippocrates. Eudoxus also considered certain curves other than the circle. There is no authority for the statement made in some old books that these were conic sections, and recent investigations have shewn that the assertion (which I repeated in the earlier editions of this book) that they were plane sections of the anchor-ring is also improbable. It seems most likely that they were tortuous curves; whatever they were, he applied them in explaining the apparent motions of the planets as seen from the earth.

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Eudoxus constructed an orrery, and wrote a treatise on practical astronomy, in which he supposed a number of moving spheres to which the sun, moon, and stars were attached, and which by their rotation produced the effects observed. In all he required twenty-seven spheres. As observations became more accurate, subsequent astronomers who accepted the theory had continually to introduce fresh spheres to make the theory agree with the facts. The work of Aratus on astronomy, which was written about 300 b.c. and is still extant, is founded on that of Eudoxus. Plato and Eudoxus were contemporaries. Among Plato’s pupils were the mathematicians Leodamas, Neocleides, Amyclas, and to their school also belonged Leon, Theudius (both of whom wrote text-books on plane geometry), Cyzicenus, Thasus, Hermotimus, Philippus, and Theaetetus. Among the pupils of Eudoxus are reckoned Menaechmus, his brother Dinostratus (who applied the quadratrix to the duplication and trisection problems), and Aristaeus. Menaechmus. Of the above-mentioned mathematicians Menaechmus requires special mention. He was born about 375 b.c., and died about 325 b.c. Probably he succeeded Eudoxus as head of the school at Cyzicus, where he acquired great reputation as a teacher of geometry, and was for that reason appointed one of the tutors of Alexander the Great. In answer to his pupil’s request to make his proofs shorter, Menaechmus made the well-known reply that though in the country there are private and even royal roads, yet in geometry there is only one road for all. Menaechmus was the first to discuss the conic sections, which were long called the Menaechmian triads. He divided them into three classes, and investigated their properties, not by taking different plane sections of a fixed cone, but by keeping his plane fixed and cutting it by different cones. He shewed that the section of a right cone by a plane perpendicular to a generator is an ellipse, if the cone be acute-angled; a parabola, if it be right-angled; and a hyperbola, if it be obtuse-angled; and he gave a mechanical construction for curves of each class. It seems almost certain that he was acquainted with the fundamental properties of these curves; but some writers think that he failed to connect them with the sections of the cone which he had discovered, and there is no doubt that he regarded the latter not as plane loci but as curves drawn on the surface of a cone. He also shewed how these curves could be used in either of the two following ways to give a solution of the problem to duplicate a cube. In

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the first of these, he pointed out that two parabolas having a common vertex, axes at right angles, and such that the latus rectum of the one is double that of the other will intersect in another point whose abscissa (or ordinate) will give a solution; for (using analysis) if the equations of the parabolas be y 2 = 2ax and x2 = ay, they intersect in a point whose abscissa is given by x3 = 2a3 . It is probable that this method was suggested by the form in which Hippocrates had cast the problem; namely, to find x and y so that a : x = x : y = y : 2a, whence we have x2 = ay and y 2 = 2ax. The second solution given by Menaechmus was as follows. Describe a parabola of latus rectum l. Next describe a rectangular hyperbola, the length of whose real axis is 4l, and having for its asymptotes the tangent at the vertex of the parabola and the axis of the parabola. Then the ordinate and the abscissa of the point of intersection of these curves are the mean proportionals between l and 2l. This is at once obvious by analysis. The curves are x2 = ly and xy = 2l2 . These cut in a point determined by x3 = 2l3 and y 3 = 4l3 . Hence l : x = x : y = y : 2l. Aristaeus and Theaetetus. Of the other members of these schools, Aristaeus and Theaetetus, whose works are entirely lost, were mathematicians of repute. We know that Aristaeus wrote on the five regular solids and on conic sections, and that Theaetetus developed the theory of incommensurable magnitudes. The only theorem we can now definitely ascribe to the latter is that given by Euclid in the ninth proposition of the tenth book of the Elements, namely, that the squares on two commensurable right lines have one to the other a ratio which a square number has to a square number (and conversely); but the squares on two incommensurable right lines have one to the other a ratio which cannot be expressed as that of a square number to a square number (and conversely). This theorem includes the results given by Theodorus.1 The contemporaries or successors of these mathematicians wrote some fresh text-books on the elements of geometry and the conic sections, introduced problems concerned with finding loci, and systematized the knowledge already acquired, but they originated no new methods of research. Aristotle. An account of the Athenian school would be incomplete if there were no mention of Aristotle, who was born at Stagira in Macedonia in 384 b.c. and died at Chalcis in Euboea in 322 b.c. Aris1

See above, p. 24.

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totle, however, deeply interested though he was in natural philosophy, was chiefly concerned with mathematics and mathematical physics as supplying illustrations of correct reasoning. A small book containing a few questions on mechanics which is sometimes attributed to him is of doubtful authority; but, though in all probability it is not his work, it is interesting, partly as shewing that the principles of mechanics were beginning to excite attention, and partly as containing the earliest known employment of letters to indicate magnitudes. The most instructive parts of the book are the dynamical proof of the parallelogram of forces for the direction of the resultant, and the statement, in effect, that if α be a force, β the mass to which it is applied, γ the distance through which it is moved, and δ the time of the motion, then α will move 12 β through 2γ in the time δ, or through γ in the time 12 δ: but the author goes on to say that it does not follow that 21 α will move β through 12 γ in the time δ, because 12 α may not be able to move β at all; for 100 men may drag a ship 100 yards, but it does not follow that one man can drag it one yard. The first part of this statement is correct and is equivalent to the statement that an impulse is proportional to the momentum produced, but the second part is wrong. The author also states the fact that what is gained in power is lost in speed, and therefore that two weights which keep a [weightless] lever in equilibrium are inversely proportional to the arms of the lever; this, he says, is the explanation why it is easier to extract teeth with a pair of pincers than with the fingers. Among other questions raised, but not answered, are why a projectile should ever stop, and why carriages with large wheels are easier to move than those with small. I ought to add that the book contains some gross blunders, and as a whole is not as able or suggestive as might be inferred from the above extracts. In fact, here as elsewhere, the Greeks did not sufficiently realise that the fundamental facts on which the mathematical treatment of mechanics must be based can be established only by carefully devised observations and experiments.

41

CHAPTER IV. the first alexandrian school.1 circ. 300 b.c.–30 b.c. The earliest attempt to found a university, as we understand the word, was made at Alexandria. Richly endowed, supplied with lecture rooms, libraries, museums, laboratories, and gardens, it became at once the intellectual metropolis of the Greek race, and remained so for a thousand years. It was particularly fortunate in producing within the first century of its existence three of the greatest mathematicians of antiquity—Euclid, Archimedes, and Apollonius. They laid down the lines on which mathematics subsequently developed, and treated it as a subject distinct from philosophy: hence the foundation of the Alexandrian Schools is rightly taken as the commencement of a new era. Thenceforward, until the destruction of the city by the Arabs in 641 a.d., the history of mathematics centres more or less round that of Alexandria; for this reason the Alexandrian Schools are commonly taken to include all Greek mathematicians of their time. The city and university of Alexandria were created under the following circumstances. Alexander the Great had ascended the throne of Macedonia in 336 b.c. at the early age of twenty, and by 332 b.c. he had conquered or subdued Greece, Asia Minor, and Egypt. Following 1

The history of the Alexandrian Schools is discussed by G. Loria in his Le Scienze Esatte nell’ Antica Grecia, Modena, 1893–1900; by Cantor, chaps. xii–xxiii; and by J. Gow in his History of Greek Mathematics, Cambridge, 1884. The subject of Greek algebra is treated by E. H. F. Nesselmann in his Die Algebra der Griechen, Berlin, 1842; see also L. Matthiessen, Grundz¨ uge der antiken und modernen Algebra der litteralen Gleichungen, Leipzig, 1878. The Greek treatment of the conic sections forms the subject of Die Lehre von den Kegelschnitten in Altertum, by H. G. Zeuthen, Copenhagen, 1886. The materials for the history of these schools have been subjected to a searching criticism by S. P. Tannery, and most of his papers are collected in his G´eom´etrie Grecque, Paris, 1887.

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the plan he adopted whenever a commanding site had been left unoccupied, he founded a new city on the Mediterranean near one mouth of the Nile; and he himself sketched out the ground-plan, and arranged for drafts of Greeks, Egyptians, and Jews to be sent to occupy it. The city was intended to be the most magnificent in the world, and, the better to secure this, its erection was left in the hands of Dinocrates, the architect of the temple of Diana at Ephesus. After Alexander’s death in 323 b.c. his empire was divided, and Egypt fell to the lot of Ptolemy, who chose Alexandria as the capital of his kingdom. A short period of confusion followed, but as soon as Ptolemy was settled on the throne, say about 306 b.c., he determined to attract, so far as he was able, learned men of all sorts to his new city; and he at once began the erection of the university buildings on a piece of ground immediately adjoining his palace. The university was ready to be opened somewhere about 300 b.c., and Ptolemy, who wished to secure for its staff the most eminent philosophers of the time, naturally turned to Athens to find them. The great library which was the central feature of the scheme was placed under Demetrius Phalereus, a distinguished Athenian, and so rapidly did it grow that within forty years it (together with the Egyptian annexe) possessed about 600,000 rolls. The mathematical department was placed under Euclid, who was thus the first, as he was one of the most famous, of the mathematicians of the Alexandrian school. It happens that contemporaneously with the foundation of this school the information on which our history is based becomes more ample and certain. Many of the works of the Alexandrian mathematicians are still extant; and we have besides an invaluable treatise by Pappus, described below, in which their best-known treatises are collated, discussed, and criticized. It curiously turns out that just as we begin to be able to speak with confidence on the subject-matter which was taught, we find that our information as to the personality of the teachers becomes vague; and we know very little of the lives of the mathematicians mentioned in this and the next chapter, even the dates at which they lived being frequently in doubt.

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The third century before Christ. Euclid.1 —This century produced three of the greatest mathematicians of antiquity, namely Euclid, Archimedes, and Apollonius. The earliest of these was Euclid. Of his life we know next to nothing, save that he was of Greek descent, and was born about 330 b.c.; he died about 275 b.c. It would appear that he was well acquainted with the Platonic geometry, but he does not seem to have read Aristotle’s works; and these facts are supposed to strengthen the tradition that he was educated at Athens. Whatever may have been his previous training and career, he proved a most successful teacher when settled at Alexandria. He impressed his own individuality on the teaching of the new university to such an extent that to his successors and almost to his contemporaries the name Euclid meant (as it does to us) the book or books he wrote, and not the man himself. Some of the medieval writers went so far as to deny his existence, and with the ingenuity of philologists they explained that the term was only a corruption of ὐκλι a key, and δις geometry. The former word was presumably derived from κλείς. I can only explain the meaning assigned to δις by the conjecture that as the Pythagoreans said that the number two symbolized a line, possibly a schoolman may have thought that it could be taken as indicative of geometry. From the meagre notices of Euclid which have come down to us we find that the saying that there is no royal road in geometry was attributed to Euclid as well as to Menaechmus; but it is an epigrammatic remark which has had many imitators. According to tradition, Euclid was noticeable for his gentleness and modesty. Of his teaching, an anecdote has been preserved. Stobaeus, who is a somewhat doubtful authority, tells us that, when a lad who had just begun geometry asked, “What do I gain by learning all this stuff?” Euclid insisted that knowledge was worth acquiring for its own sake, but made his slave give the boy some coppers, “since,” said he, “he must make a profit out of what he learns.” 1

Besides Loria, book ii, chap. i; Cantor, chaps. xii, xiii; and Gow, pp. 72–86, 195– 221; see the articles Eucleides by A. De Morgan in Smith’s Dictionary of Greek and Roman Biography, London, 1849; the article on Irrational Quantity by A. De Morgan in the Penny Cyclopaedia, London, 1839; Litterargeschichtliche Studien u ¨ber Euklid, by J. L. Heiberg, Leipzig, 1882; and above all Euclid’s Elements, translated with an introduction and commentary by T. L. Heath, 3 volumes, Cambridge, 1908. The latest complete edition of all Euclid’s works is that by J. L. Heiberg and H. Menge, Leipzig, 1883–96.

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Euclid was the author of several works, but his reputation rests mainly on his Elements. This treatise contains a systematic exposition of the leading propositions of elementary metrical geometry (exclusive of conic sections) and of the theory of numbers. It was at once adopted by the Greeks as the standard text-book on the elements of pure mathematics, and it is probable that it was written for that purpose and not as a philosophical attempt to shew that the results of geometry and arithmetic are necessary truths. The modern text1 is founded on an edition or commentary prepared by Theon, the father of Hypatia (circ. 380 a.d.). There is at the Vatican a copy (circ. 1000 a.d.) of an older text, and we have besides quotations from the work and references to it by numerous writers of various dates. From these sources we gather that the definitions, axioms, and postulates were rearranged and slightly altered by subsequent editors, but that the propositions themselves are substantially as Euclid wrote them. As to the matter of the work. The geometrical part is to a large extent a compilation from the works of previous writers. Thus the substance of books i and ii (except perhaps the treatment of parallels) is probably due to Pythagoras; that of book iii to Hippocrates; that of book v to Eudoxus; and the bulk of books iv, vi, xi, and xii to the later Pythagorean or Athenian schools. But this material was rearranged, obvious deductions were omitted (for instance, the proposition that the perpendiculars from the angular points of a triangle on the opposite sides meet in a point was cut out), and in some cases new proofs substituted. Book X, which deals with irrational magnitudes, may be founded on the lost book of Theaetetus; but probably much of it is original, for Proclus says that while Euclid arranged the propositions of Eudoxus he completed many of those of Theaetetus. The whole was presented as a complete and consistent body of theorems. The form in which the propositions are presented, consisting of enunciation, statement, construction, proof, and conclusion, is due to Euclid: so also is the synthetical character of the work, each proof being written out as a logically correct train of reasoning but without any clue to the method by which it was obtained. 1

Most of the modern text-books in English are founded on Simson’s edition, issued in 1758. Robert Simson, who was born in 1687 and died in 1768, was professor of mathematics at the University of Glasgow, and left several valuable works on ancient geometry.

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The defects of Euclid’s Elements as a text-book of geometry have been often stated; the most prominent are these. (i) The definitions and axioms contain many assumptions which are not obvious, and in particular the postulate or axiom about parallel lines is not self-evident.1 (ii) No explanation is given as to the reason why the proofs take the form in which they are presented, that is, the synthetical proof is given but not the analysis by which it was obtained. (iii) There is no attempt made to generalize the results arrived at; for instance, the idea of an angle is never extended so as to cover the case where it is equal to or greater than two right angles: the second half of the thirty-third proposition in the sixth book, as now printed, appears to be an exception, but it is due to Theon and not to Euclid. (iv) The principle of superposition as a method of proof might be used more frequently with advantage. (v) The classification is imperfect. And (vi) the work is unnecessarily long and verbose. Some of those objections do not apply to certain of the recent school editions of the Elements. On the other hand, the propositions in Euclid are arranged so as to form a chain of geometrical reasoning, proceeding from certain almost obvious assumptions by easy steps to results of considerable complexity. The demonstrations are rigorous, often elegant, and not too difficult for a beginner. Lastly, nearly all the elementary metrical (as opposed to the graphical) properties of space are investigated, while the fact that for two thousand years it was the usual text-book on the subject raises a strong presumption that it is not unsuitable for the purpose. On the Continent rather more than a century ago, Euclid was generally superseded by other text-books. In England determined efforts have lately been made with the same purpose, and numerous other works on elementary geometry have been produced in the last decade. The change is too recent to enable us to say definitely what its effect may be. But as far as I can judge, boys who have learnt their geometry on the new system know more facts, but have missed the mental and logical training which was inseparable from a judicious study of Euclid’s treatise. I do not think that all the objections above stated can fairly be urged against Euclid himself. He published a collection of problems, generally known as the Δεδομένα or Data. This contains 95 illustrations of the kind of deductions which frequently have to be made in analysis; 1

We know, from the researches of Lobatschewsky and Riemann, that it is incapable of proof.

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such as that, if one of the data of the problem under consideration be that one angle of some triangle in the figure is constant, then it is legitimate to conclude that the ratio of the area of the rectangle under the sides containing the angle to the area of the triangle is known [prop. 66]. Pappus says that the work was written for those “who wish to acquire the power of solving problems.” It is in fact a gradual series of exercises in geometrical analysis. In short the Elements gave the principal results, and were intended to serve as a training in the science of reasoning, while the Data were intended to develop originality. Euclid also wrote a work called Περὶ Διαιρέσεων or De Divisionibus, known to us only through an Arabic translation which may be itself imperfect.1 This is a collection of 36 problems on the division of areas into parts which bear to one another a given ratio. It is not unlikely that this was only one of several such collections of examples—possibly including the Fallacies and the Porisms—but even by itself it shews that the value of exercises and riders was fully recognized by Euclid. I may here add a suggestion made by De Morgan, whose comments on Euclid’s writings were notably ingenious and informing. From internal evidence he thought it likely that the Elements were written towards the close of Euclid’s life, and that their present form represents the first draft of the proposed work, which, with the exception of the tenth book, Euclid did not live to revise. This opinion is generally discredited, and there is no extrinsic evidence to support it. The geometrical parts of the Elements are so well known that I need do no more than allude to them. Euclid admitted only those constructions which could be made by the use of a ruler and compasses.2 He also excluded practical work and hypothetical constructions. The first four books and book vi deal with plane geometry; the theory of proportion (of any magnitudes) is discussed in book v; and books xi and xii treat of solid geometry. On the hypothesis that the Elements are the first draft of Euclid’s proposed work, it is possible that book xiii 1

R. C. Archibald, Euclid’s Book on Divisions, Cambridge, 1915. The ruler must be of unlimited length and not graduated; the compasses also must be capable of being opened as wide as is desired. Lorenzo Mascheroni (who was born at Castagneta on May 14, 1750, and died at Paris on July 30, 1800) set himself the task to obtain by means of constructions made only with a pair of compasses as many Euclidean results as possible. Mascheroni’s treatise on the geometry of the compass, which was published at Pavia in 1795, is a curious tour de force: he was professor first at Bergamo and afterwards at Pavia, and left numerous minor works. Similar limitations have been proposed by other writers. 2

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is a sort of appendix containing some additional propositions which would have been put ultimately in one or other of the earlier books. Thus, as mentioned above, the first five propositions which deal with a line cut in golden section might be added to the second book. The next seven propositions are concerned with the relations between certain incommensurable lines in plane figures (such as the radius of a circle and the sides of an inscribed regular triangle, pentagon, hexagon, and decagon) which are treated by the methods of the tenth book and as an illustration of them. Constructions of the five regular solids are discussed in the last six propositions, and it seems probable that Euclid and his contemporaries attached great importance to this group of problems. Bretschneider inclined to think that the thirteenth book is a summary of part of the lost work of Aristaeus: but the illustrations of the methods of the tenth book are due most probably to Theaetetus. Books vii, viii, ix, and x of the Elements are given up to the theory of numbers. The mere art of calculation or λογιστική was taught to boys when quite young, it was stigmatized by Plato as childish, and never received much attention from Greek mathematicians; nor was it regarded as forming part of a course of mathematics. We do not know how it was taught, but the abacus certainly played a prominent part in it. The scientific treatment of numbers was called ἀριθμητική, which I have here generally translated as the science of numbers. It had special reference to ratio, proportion, and the theory of numbers. It is with this alone that most of the extant Greek works deal. In discussing Euclid’s arrangement of the subject, we must therefore bear in mind that those who attended his lectures were already familiar with the art of calculation. The system of numeration adopted by the Greeks is described later,1 but it was so clumsy that it rendered the scientific treatment of numbers much more difficult than that of geometry; hence Euclid commenced his mathematical course with plane geometry. At the same time it must be observed that the results of the second book, though geometrical in form, are capable of expression in algebraical language, and the fact that numbers could be represented by lines was probably insisted on at an early stage, and illustrated by concrete examples. This graphical method of using lines to represent numbers possesses the obvious advantage of leading to proofs which are true for all numbers, rational or irrational. It will be noticed that among other propositions in the second book we get geometrical proofs 1

See below, chapter vii.

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of the distributive and commutative laws, of rules for multiplication, and finally geometrical solutions of the equations a(a − x) = x2 , that is x2 +ax−a2 = 0 (Euc. ii, 11), and x2 −ab = 0 (Euc. ii, √14): the solution of the first of these equations is given in the form a2 + ( 12 a)2 − 12 a. The solutions of the equations ax2 − bx + c = 0 and ax2 + bx − c = 0 are given later in Euc. vi, 28 and vi, 29; the cases when a = 1 can be deduced from the identities proved in Euc. ii, 5 and 6, but it is doubtful if Euclid recognized this. The results of the fifth book, in which the theory of proportion is considered, apply to any magnitudes, and therefore are true of numbers as well as of geometrical magnitudes. In the opinion of many writers this is the most satisfactory way of treating the theory of proportion on a scientific basis; and it was used by Euclid as the foundation on which he built the theory of numbers. The theory of proportion given in this book is believed to be due to Eudoxus. The treatment of the same subject in the seventh book is less elegant, and is supposed to be a reproduction of the Pythagorean teaching. This double discussion of proportion is, as far as it goes, in favour of the conjecture that Euclid did not live to revise the work. In books vii, viii, and ix Euclid discusses the theory of rational numbers. He commences the seventh book with some definitions founded on the Pythagorean notation. In propositions 1 to 3 he shews that if, in the usual process for finding the greatest common measure of two numbers, the last divisor be unity, the numbers must be prime; and he thence deduces the rule for finding their G.C.M. Propositions 4 to 22 include the theory of fractions, which he bases on the theory of proportion; among other results he shews that ab = ba [prop. 16]. In propositions 23 to 34 he treats of prime numbers, giving many of the theorems in modern text-books on algebra. In propositions 35 to 41 he discusses the least common multiple of numbers, and some miscellaneous problems. The eighth book is chiefly devoted to numbers in continued proportion, that is, in a geometrical progression; and the cases where one or more is a product, square, or cube are specially considered. In the ninth book Euclid continues the discussion of geometrical progressions, and in proposition 35 he enunciates the rule for the summation of a series of n terms, though the proof is given only for the case where n is equal to 4. He also develops the theory of primes, shews that the number of primes is infinite [prop. 20], and discusses the properties

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of odd and even numbers. He concludes by shewing that a number of the form 2n−1 (2n − 1), where 2n − 1 is a prime, is a “perfect” number [prop. 36]. In the tenth book Euclid deals with certain irrational magnitudes; and, since the Greeks possessed no symbolism for surds, he was forced to adopt a geometrical representation. Propositions 1 to 21 deal generally with incommensurable magnitudes. The rest of the book, namely, propositions 22 to 117, is devoted to the discussion √√ √of every possible variety of lines which can be represented by ( a ± b), where a and b denote commensurable lines. There are twenty-five species of such lines, and that Euclid could detect and classify them all is in the opinion of so competent an authority as Nesselmann the most striking illustration of his genius. No further advance in the theory of incommensurable magnitudes was made until the subject was taken up by Leonardo and Cardan after an interval of more than a thousand years. In the last proposition of the tenth book [prop. 117] the side and diagonal of a square are proved to be incommensurable. The proof is so short and easy that I may quote it. If possible let the side be to the diagonal in a commensurable ratio, namely, that of two integers, a and b. Suppose this ratio reduced to its lowest terms so that a and b have no common divisor other than unity, that is, they are prime to one another. Then (by Euc. i, 47) b2 = 2a2 ; therefore b2 is an even number; therefore b is an even number; hence, since a is prime to b, a must be an odd number. Again, since it has been shewn that b is an even number, b may be represented by 2n; therefore (2n)2 = 2a2 ; therefore a2 = 2n2 ; therefore a2 is an even number; therefore a is an even number. Thus the same number a must be both odd and even, which is absurd; therefore the side and diagonal are incommensurable. Hankel believes that this proof was due to Pythagoras, and this is not unlikely. This proposition is also proved in another way in Euc. x, 9, and for this and other reasons it is now usually believed to be an interpolation by some commentator on the Elements. In addition to the Elements and the two collections of riders above mentioned (which are extant) Euclid wrote the following books on geometry: (i) an elementary treatise on conic sections in four books; (ii) a book on surface loci, probably confined to curves on the cone and cylinder; (iii) a collection of geometrical fallacies, which were to be used as exercises in the detection of errors; and (iv) a treatise on porisms arranged in three books. All of these are lost, but the work on porisms was discussed at such length by Pappus, that some writers have thought

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it possible to restore it. In particular, Chasles in 1860 published what he considered to be substantially a reproduction of it. In this will be found the conceptions of cross ratios and projection, and those ideas of modern geometry which were used so extensively by Chasles and other writers of the nineteenth century. It should be realized, however, that the statements of the classical writers concerning this book are either very brief or have come to us only in a mutilated form, and De Morgan frankly says that he found them unintelligible, an opinion in which most of those who read them will, I think, concur. Euclid published a book on optics, treated geometrically, which contains 61 propositions founded on 12 assumptions. It commences with the assumption that objects are seen by rays emitted from the eye in straight lines, “for if light proceeded from the object we should not, as we often do, fail to perceive a needle on the floor.” A work called Catoptrica is also attributed to him by some of the older writers; the text is corrupt and the authorship doubtful; it consists of 31 propositions dealing with reflexions in plane, convex, and concave mirrors. The geometry of both books is Euclidean in form. Euclid has been credited with an ingenious demonstration1 of the principle of the lever, but its authenticity is doubtful. He also wrote the Phaenomena, a treatise on geometrical astronomy. It contains references to the work of Autolycus2 and to some book on spherical geometry by an unknown writer. Pappus asserts that Euclid also composed a book on the elements of music: this may refer to the Sectio Canonis, which is by Euclid, and deals with musical intervals. To these works I may add the following little problem, which occurs in the Palatine Anthology and is attributed by tradition to Euclid. “A mule and a donkey were going to market laden with wheat. The mule said, ‘If you gave me one measure I should carry twice as much as you, but if I gave you one we should bear equal burdens.’ Tell me, learned geometrician, what were their burdens.” It is impossible to say whether the question is due to Euclid, but there is nothing improbable in the suggestion. It will be noticed that Euclid dealt only with magnitudes, and did 1 It is given (from the Arabic) by F. Woepcke in the Journal Asiatique, series 4, vol. xviii, October 1851, pp. 225–232. 2 Autolycus lived at Pitane in Aeolis and flourished about 330 b.c. His two works on astronomy, containing 43 propositions, are said to be the oldest extant Greek mathematical treatises. They exist in manuscript at Oxford. They were edited, with a Latin translation, by F. Hultsch, Leipzig, 1885.

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not concern himself with their numerical measures, but it would seem from the works of Aristarchus and Archimedes that this was not the case with all the Greek mathematicians of that time. As one of the works of the former is extant it will serve as another illustration of Greek mathematics of this period. Aristarchus. Aristarchus of Samos, born in 310 b.c. and died in 250 b.c., was an astronomer rather than a mathematician. He asserted, at any rate as a working hypothesis, that the sun was the centre of the universe, and that the earth revolved round the sun. This view, in spite of the simple explanation it afforded of various phenomena, was generally rejected by his contemporaries. But his propositions1 on the measurement of the sizes and distances of the sun and moon were accurate in principle, and his results were accepted by Archimedes in his Ψαμμίτης, mentioned below, as approximately correct. There are 19 theorems, of which I select the seventh as a typical illustration, because it shews the way in which the Greeks evaded the difficulty of finding the numerical value of surds. Aristarchus observed the angular distance between the moon when dichotomized and the sun, and found it to be twenty-nine thirtieths of a right angle. It is actually about 89◦ 210 , but of course his instruments were of the roughest description. He then proceeded to shew that the distance of the sun is greater than eighteen and less than twenty times the distance of the moon in the following manner. Let S be the sun, E the earth, and M the moon. Then when the moon is dichotomized, that is, when the bright part which we see is exactly a half-circle, the angle between M S and M E is a right angle. With E as centre, and radii ES and EM describe circles, as in the figure below. Draw EA perpendicular to ES. Draw EF bisecting the angle AES, and EG bisecting the angle AEF , as in the figure. Let 1 th EM (produced) cut AF in H. The angle AEM is by hypothesis 30 of a right angle. Hence we have 1 angle AEG : angle AEH = 14 rt. ∠ : 30 rt. ∠ = 15 : 2, ∴ AG : AH [ = tan AEG : tan AEH] > 15 : 2.

1

(α)

Περὶ μεγέθων καὶ ἀποστημάτων `Ηλίου καὶ Σελήνης, edited by E. Nizze, Stralsund, 1856. Latin translations were issued by F. Commandino in 1572 and by J. Wallis in 1688; and a French translation was published by F. d’Urban in 1810 and 1823.

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G

H

A

M

S

E

Again F G2 : AG2 = EF 2 : EA2 (Euc. vi, 3) = 2 : 1 (Euc. i, 47), ∴ F G2 : AG2 > 49 : 25, ∴ F G : AG > 7 : 5, ∴ AF : AG > 12 : 5, ∴ AE : AG > 12 : 5.

(β)

Compounding the ratios (α) and (β), we have AE : AH > 18 : 1. But the triangles EM S and EAH are similar, ∴ ES : EM > 18 : 1. I will leave the second half of the proposition to amuse any reader who may care to prove it: the analysis is straightforward. In a somewhat similar way Aristarchus found the ratio of the radii of the sun, earth, and moon. We know very little of Conon and Dositheus, the immediate successors of Euclid at Alexandria, or of their contemporaries Zeuxippus

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and Nicoteles, who most likely also lectured there, except that Archimedes, who was a student at Alexandria probably shortly after Euclid’s death, had a high opinion of their ability and corresponded with the three first mentioned. Their work and reputation has been completely overshadowed by that of Archimedes. Archimedes.1 Archimedes, who probably was related to the royal family at Syracuse, was born there in 287 b.c. and died in 212 b.c. He went to the university of Alexandria and attended the lectures of Conon, but, as soon as he had finished his studies, returned to Sicily where he passed the remainder of his life. He took no part in public affairs, but his mechanical ingenuity was astonishing, and, on any difficulties which could be overcome by material means arising, his advice was generally asked by the government. Archimedes, like Plato, held that it was undesirable for a philosopher to seek to apply the results of science to any practical use; but in fact he did introduce a large number of new inventions. The stories of the detection of the fraudulent goldsmith and of the use of burningglasses to destroy the ships of the Roman blockading squadron will recur to most readers. Perhaps it is not as well known that Hiero, who had built a ship so large that he could not launch it off the slips, applied to Archimedes. The difficulty was overcome by means of an apparatus of cogwheels worked by an endless screw, but we are not told exactly how the machine was used. It is said that it was on this occasion, in acknowledging the compliments of Hiero, that Archimedes made the well-known remark that had he but a fixed fulcrum he could move the earth. Most mathematicians are aware that the Archimedean screw was another of his inventions. It consists of a tube, open at both ends, and bent into the form of a spiral like a corkscrew. If one end be immersed in water, and the axis of the instrument (i.e. the axis of the cylinder on the surface of which the tube lies) be inclined to the vertical at a sufficiently big angle, and the instrument turned round it, the water 1

Besides Loria, book ii, chap. iii, Cantor, chaps. xiv, xv, and Gow, pp. 221– 244, see Quaestiones Archimedeae, by J. L. Heiberg, Copenhagen, 1879; and Marie, vol. i, pp. 81–134. The best editions of the extant works of Archimedes are those by J. L. Heiberg, in 3 vols., Leipzig, 1880–81, and by Sir Thomas L. Heath, Cambridge, 1897. In 1906 a manuscript, previously unknown, was discovered at Constantinople, containing propositions on hydrostatics and on methods; see Eine neue Schrift des Archimedes, by J. L. Heiberg and H. G. Zeuthen, Leipzig, 1907, and the Method of Archimedes, by Sir Thomas L. Heath, Cambridge, 1912.

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will flow along the tube and out at the other end. In order that it may work, the inclination of the axis of the instrument to the vertical must be greater than the pitch of the screw. It was used in Egypt to drain the fields after an inundation of the Nile, and was also frequently applied to take water out of the hold of a ship. The story that Archimedes set fire to the Roman ships by means of burning-glasses and concave mirrors is not mentioned till some centuries after his death, and is generally rejected. The mirror of Archimedes is said to have been made in the form of a hexagon surrounded by rings of polygons; and Buffon1 in 1747 contrived, by the use of a single composite mirror made on this model, to set fire to wood at a distance of 150 feet, and to melt lead at a distance of 140 feet. This was in April and as far north as Paris, so in a Sicilian summer the use of several such mirrors might be a serious annoyance to a blockading fleet, if the ships were sufficiently near. It is perhaps worth mentioning that a similar device is said to have been used in the defence of Constantinople in 514 a.d., and is alluded to by writers who either were present at the siege or obtained their information from those who were engaged in it. But whatever be the truth as to this story, there is no doubt that Archimedes devised the catapults which kept the Romans, who were then besieging Syracuse, at bay for a considerable time. These were constructed so that the range could be made either short or long at pleasure, and so that they could be discharged through a small loophole without exposing the artillery-men to the fire of the enemy. So effective did they prove that the siege was turned into a blockade, and three years elapsed before the town was taken. Archimedes was killed during the sack of the city which followed its capture, in spite of the orders, given by the consul Marcellus who was in command of the Romans, that his house and life should be spared. It is said that a soldier entered his study while he was regarding a geometrical diagram drawn in sand on the floor, which was the usual way of drawing figures in classical times. Archimedes told him to get off the diagram, and not spoil it. The soldier, feeling insulted at having orders given to him and ignorant of who the old man was, killed him. According to another and more probable account, the cupidity of the troops was excited by seeing his instruments, constructed of polished brass which they supposed to be made of gold. 1

101.

See M´emoires de l’acad´emie royale des sciences for 1747, Paris, 1752, pp. 82–

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The Romans erected a splendid tomb to Archimedes, on which was engraved (in accordance with a wish he had expressed) the figure of a sphere inscribed in a cylinder, in commemoration of the proof he had given that the volume of a sphere was equal to two-thirds that of the circumscribing right cylinder, and its surface to four times the area of a great circle. Cicero1 gives a charming account of his efforts (which were successful) to rediscover the tomb in 75 b.c. It is difficult to explain in a concise form the works or discoveries of Archimedes, partly because he wrote on nearly all the mathematical subjects then known, and partly because his writings are contained in a series of disconnected monographs. Thus, while Euclid aimed at producing systematic treatises which could be understood by all students who had attained a certain level of education, Archimedes wrote a number of brilliant essays addressed chiefly to the most educated mathematicians of the day. The work for which he is perhaps now best known is his treatment of the mechanics of solids and fluids; but he and his contemporaries esteemed his geometrical discoveries of the quadrature of a parabolic area and of a spherical surface, and his rule for finding the volume of a sphere as more remarkable; while at a somewhat later time his numerous mechanical inventions excited most attention. (i) On plane geometry the extant works of Archimedes are three in number, namely, (a) the Measure of the Circle, (b) the Quadrature of the Parabola, and (c) one on Spirals. (a) The Measure of the Circle contains three propositions. In the first proposition Archimedes proves that the area is the same as that of a right-angled triangle whose sides are equal respectively to the radius a and the circumference of the circle, i.e. the area is equal to 12 a(2πa). In the second proposition he shows that πa2 : (2a)2 = 11 : 14 very nearly; and next, in the third proposition, that π is less than 3 71 and greater than 3 10 . These theorems are of course proved geometrically. 71 To demonstrate the two latter propositions, he inscribes in and circumscribes about a circle regular polygons of ninety-six sides, calculates their perimeters, and then assumes the circumference of the circle to lie between them: this leads to the result 6336/2017 14 < π < 14688/4673 12 , from which he deduces the limits given above. It would seem from the proof that he had some (at present unknown) method of extracting the square roots of numbers approximately. The table which he formed of the numerical values of the chords of a circle is essentially a table of 1

See his Tusculanarum Disputationum, v. 23.

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natural sines, and may have suggested the subsequent work on these lines of Hipparchus and Ptolemy. (b) The Quadrature of the Parabola contains twenty-four propositions. Archimedes begins this work, which was sent to Dositheus, by establishing some properties of conics [props. 1–5]. He then states correctly the area cut off from a parabola by any chord, and gives a proof which rests on a preliminary mechanical experiment of the ratio of areas which balance when suspended from the arms of a lever [props. 6–17]; and, lastly, he gives a geometrical demonstration of this result [props. 18–24]. The latter is, of course, based on the method of exhaustions, but for brevity I will, in quoting it, use the method of limits. P

V

M

Q

Let the area of the parabola (see figure above) be bounded by the chord P Q. Draw V M the diameter to the chord P Q, then (by a previous proposition), V is more remote from P Q than any other point in the arc P V Q. Let the area of the triangle P V Q be denoted by 4. In the segments bounded by V P and V Q inscribe triangles in the same way as the triangle P V Q was inscribed in the given segment. Each of these triangles is (by a previous proposition of his) equal to 81 4, and their sum is therefore 41 4. Similarly in the four segments left inscribe

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1 triangles; their sum will be 16 4. Proceeding in this way the area of the given segment is shown to be equal to the limit of

4+

4 4 4 + + ··· + n + ··· , 4 16 4

when n is indefinitely large. The problem is therefore reduced to finding the sum of a geometrical series. This he effects as follows. Let A, B, C, . . . , J, K be a series of magnitudes such that each is one-fourth of that which precedes it. Take magnitudes b, c, . . . , k equal respectively to 13 B, 13 C, . . . , 13 K. Then B + b = 13 A,

C + c = 31 B,

...,

K + k = 13 J.

Hence (B + C + . . . + K) + (b + c + . . . + k) = 13 (A + B + . . . + J); but, by hypothesis, (b + c + . . . + j + k) = 31 (B + C + . . . + J) + 13 K; ∴ (B + C + . . . + K) + 13 K = 13 A. ∴ A + B + C + . . . + K = 34 A − 13 K. Hence the sum of these magnitudes exceeds four times the third of the largest of them by one-third of the smallest of them. Returning now to the problem of the quadrature of the parabola A stands for ∆, and ultimately K is indefinitely small; therefore the area of the parabolic segment is four-thirds that of the triangle P V Q, or two-thirds that of a rectangle whose base is P Q and altitude the distance of V from P Q. While discussing the question of quadratures it may be added that in the fifth and sixth propositions of his work on conoids and spheroids he determined the area of an ellipse. (c) The work on Spirals contains twenty-eight propositions on the properties of the curve now known as the spiral of Archimedes. It was sent to Dositheus at Alexandria accompanied by a letter, from which it appears that Archimedes had previously sent a note of his results to Conon, who had died before he had been able to prove them. The spiral is defined by saying that the vectorial angle and radius vector both increase uniformly, hence its equation is r = cθ. Archimedes finds most of its properties, and determines the area inclosed between the curve and two radii vectores. This he does (in effect) by saying, in the language of the infinitesimal calculus, that an element of area is > 21 r2 dθ and < 12 (r + dr)2 dθ: to effect the sum of the elementary

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areas he gives two lemmas in which he sums (geometrically) the series a2 + (2a)2 + (3a)2 + . . . + (na)2 [prop. 10], and a + 2a + 3a + . . . + na [prop. 11]. (d ) In addition to these he wrote a small treatise on geometrical methods, and works on parallel lines, triangles, the properties of rightangled triangles, data, the heptagon inscribed in a circle, and systems of circles touching one another ; possibly he wrote others too. These are all lost, but it is probable that fragments of four of the propositions in the last-mentioned work are preserved in a Latin translation from an Arabic manuscript entitled Lemmas of Archimedes. (ii) On geometry of three dimensions the extant works of Archimedes are two in number, namely (a), the Sphere and Cylinder, and (b) Conoids and Spheroids. (a) The Sphere and Cylinder contains sixty propositions arranged in two books. Archimedes sent this like so many of his works to Dositheus at Alexandria; but he seems to have played a practical joke on his friends there, for he purposely misstated some of his results “to deceive those vain geometricians who say they have found everything, but never give their proofs, and sometimes claim that they have discovered what is impossible.” He regarded this work as his masterpiece. It is too long for me to give an analysis of its contents, but I remark in passing that in it he finds expressions for the surface and volume of a pyramid, of a cone, and of a sphere, as well as of the figures produced by the revolution of polygons inscribed in a circle about a diameter of the circle. There are several other propositions on areas and volumes of which perhaps the most striking is the tenth proposition of the second book, namely, that “of all spherical segments whose surfaces are equal the hemisphere has the greatest volume.” In the second proposition of the second book he enunciates the remarkable theorem that a line of length a can be divided so that a − x : b = 4a2 : 9x2 (where b is a given length), only if b be less than 31 a; that is to say, the cubic equation x3 − ax2 + 49 a2 b = 0 can have a real and positive root only if a be greater than 3b. This proposition was required to complete his solution of the problem to divide a given sphere by a plane so that the volumes of the segments should be in a given ratio. One very simple cubic equation occurs in the Arithmetic of Diophantus, but with that exception no such equation appears again in the history of European mathematics for more than a thousand years. (b) The Conoids and Spheroids contains forty propositions on quadrics of revolution (sent to Dositheus in Alexandria) mostly concerned

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with an investigation of their volumes. (c) Archimedes also wrote a treatise on certain semi-regular polyhedrons, that is, solids contained by regular but dissimilar polygons. This is lost, but references to it are given by Pappus. (iii) On arithmetic Archimedes wrote two papers. One (addressed to Zeuxippus) was on the principles of numeration; this is now lost. The other (addressed to Gelon) was called Ψαμμίτης (the sand-reckoner ), and in this he meets an objection which had been urged against his first paper. The object of the first paper had been to suggest a convenient system by which numbers of any magnitude could be represented; and it would seem that some philosophers at Syracuse had doubted whether the system was practicable. Archimedes says people talk of the sand on the Sicilian shore as something beyond the power of calculation, but he can estimate it; and, further, he will illustrate the power of his method by finding a superior limit to the number of grains of sand which would fill the whole universe, i.e. a sphere whose centre is the earth, and radius the distance of the sun. He begins by saying that in ordinary Greek nomenclature it was only possible to express numbers from 1 up to 108 : these are expressed in what he says he may call units of the first order. If 108 be termed a unit of the second order, any number from 108 to 1016 can be expressed as so many units of the second order plus so many units of the first order. If 1016 be a unit of the third order any number up to 1024 can be then expressed, and so on. Assuming that 1 th 10,000 grains of sand occupy a sphere whose radius is not less than 80 of a finger-breadth, and that the diameter of the universe is not greater than 1010 stadia, he finds that the number of grains of sand required to fill the solar universe is less than 1051 . Probably this system of numeration was suggested merely as a scientific curiosity. The Greek system of numeration with which we are acquainted had been only recently introduced, most likely at Alexandria, and was sufficient for all the purposes for which the Greeks then required numbers; and Archimedes used that system in all his papers. On the other hand, it has been conjectured that Archimedes and Apollonius had some symbolism based on the decimal system for their own investigations, and it is possible that it was the one here sketched out. The units suggested by Archimedes form a geometrical progression, having 108 for the radix. He incidentally adds that it will be convenient to remember that the product of the mth and nth terms of a geometrical progression, whose first term is unity, is equal to the (m + n)th

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term of the series, that is, that rm × rn = rm+n . To these two arithmetical papers I may add the following celebrated problem1 which he sent to the Alexandrian mathematicians. The sun had a herd of bulls and cows, all of which were either white, grey, dun, or piebald: the number of piebald bulls was less than the number of white bulls by 5/6ths of the number of grey bulls, it was less than the number of grey bulls by 9/20ths of the number of dun bulls, and it was less than the number of dun bulls by 13/42nds of the number of white bulls; the number of white cows was 7/12ths of the number of grey cattle (bulls and cows), the number of grey cows was 9/20ths of the number of dun cattle, the number of dun cows was 11/30ths of the number of piebald cattle, and the number of piebald cows was 13/42nds of the number of white cattle. The problem was to find the composition of the herd. The problem is indeterminate, but the solution in lowest integers is white bulls, . . . . . grey bulls, . . . . . . . dun bulls, . . . . . . . piebald bulls, . . . .

10,366,482; 7,460,514; 7,358,060; 4,149,387;

white cows,. . . . . . grey cows, . . . . . . . dun cows, . . . . . . . piebald cows, . . . .

7,206,360; 4,893,246; 3,515,820; 5,439,213.

In the classical solution, attributed to Archimedes, these numbers are multiplied by 80. Nesselmann believes, from internal evidence, that the problem has been falsely attributed to Archimedes. It certainly is unlike his extant work, but it was attributed to him among the ancients, and is generally thought to be genuine, though possibly it has come down to us in a modified form. It is in verse, and a later copyist has added the additional conditions that the sum of the white and grey bulls shall be a square number, and the sum of the piebald and dun bulls a triangular number. It is perhaps worthy of note that in the enunciation the fractions are represented as a sum of fractions whose numerators are unity: thus Archimedes wrote 1/7+1/6 instead of 13/42, in the same way as Ahmes would have done. (iv) On mechanics the extant works of Archimedes are two in number, namely, (a) his Mechanics, and (c) his Hydrostatics. 1

See a memoir by B. Krumbiegel and A. Amthor, Zeitschrift f¨ ur Mathematik, Abhandlungen zur Geschichte der Mathematik, Leipzig, vol. xxv, 1880, pp. 121–136, 153–171.

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(a) The Mechanics is a work on statics with special reference to the equilibrium of plane laminas and to properties of their centres of gravity; it consists of twenty-five propositions in two books. In the first part of book i, most of the elementary properties of the centre of gravity are proved [props. 1–8]; and in the remainder of book i, [props. 9– 15] and in book ii the centres of gravity of a variety of plane areas, such as parallelograms, triangles, trapeziums, and parabolic areas are determined. As an illustration of the influence of Archimedes on the history of mathematics, I may mention that the science of statics rested on his theory of the lever until 1586, when Stevinus published his treatise on statics. His reasoning is sufficiently illustrated by an outline of his proof for the case of two weights, P and Q, placed at their centres of gravity, A and B, on a weightless bar AB. He wants to shew that the centre of gravity of P and Q is at a point O on the bar such that P.OA = Q.OB. L

H A

K O

B

On the line AB (produced if necessary) take points H and K, so that HB = BK = AO; and a point L so that LA = OB. It follows that LH will be bisected at A, HK at B, and LK at O; also LH : HK = AH : HB = OB : AO = P : Q. Hence, by a previous proposition, we may consider that the effect of P is the same as that of a heavy uniform bar LH of weight P , and the effect of Q is the same as that of a similar heavy uniform bar HK of weight Q. Hence the effect of the weights is the same as that of a heavy uniform bar LK. But the centre of gravity of such a bar is at its middle point O. (b) Archimedes also wrote a treatise on levers and perhaps, on all the mechanical machines. The book is lost, but we know from Pappus that it contained a discussion of how a given weight could be moved with a given power. It was in this work probably that Archimedes discussed the theory of a certain compound pulley consisting of three or more simple pulleys which he had invented, and which was used in some public works in Syracuse. It is well known1 that he boasted that, if he had but a fixed fulcrum, he could move the whole earth; and a 1

See above, p. 53.

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commentator of later date says that he added he would do it by using a compound pulley. (c) His work on floating bodies contains nineteen propositions in two books, and was the first attempt to apply mathematical reasoning to hydrostatics. The story of the manner in which his attention was directed to the subject is told by Vitruvius. Hiero, the king of Syracuse, had given some gold to a goldsmith to make into a crown. The crown was delivered, made up, and of the proper weight, but it was suspected that the workman had appropriated some of the gold, replacing it by an equal weight of silver. Archimedes was thereupon consulted. Shortly afterwards, when in the public baths, he noticed that his body was pressed upwards by a force which increased the more completely he was immersed in the water. Recognising the value of the observation, he rushed out, just as he was, and ran home through the streets, shouting εὕρηκα, εὕρηκα, “I have found it, I have found it.” There (to follow a later account) on making accurate experiments he found that when equal weights of gold and silver were weighed in water they no longer appeared equal: each seemed lighter than before by the weight of the water it displaced, and as the silver was more bulky than the gold its weight was more diminished. Hence, if on a balance he weighed the crown against an equal weight of gold and then immersed the whole in water, the gold would outweigh the crown if any silver had been used in its construction. Tradition says that the goldsmith was found to be fraudulent. Archimedes began the work by proving that the surface of a fluid at rest is spherical, the centre of the sphere being at the centre of the earth. He then proved that the pressure of the fluid on a body, wholly or partially immersed, is equal to the weight of the fluid displaced; and thence found the position of equilibrium of a floating body, which he illustrated by spherical segments and paraboloids of revolution floating on a fluid. Some of the latter problems involve geometrical reasoning of considerable complexity. The following is a fair specimen of the questions considered. A solid in the shape of a paraboloid of revolution of height h and latus rectum 4a floats in water, with its vertex immersed and its base wholly above the surface. If equilibrium be possible when the axis is not vertical, then the density of the body must be less than (h − 3a)2 /h3 [book ii, prop. 4]. When it is recollected that Archimedes was unacquainted with trigonometry or analytical geometry, the fact that he could discover and prove a proposition such as that just quoted will serve as an illustration

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of his powers of analysis. It will be noticed that the mechanical investigations of Archimedes were concerned with statics. It may be added that though the Greeks attacked a few problems in dynamics, they did it with but indifferent success: some of their remarks were acute, but they did not sufficiently realise that the fundamental facts on which the theory must be based can be established only by carefully devised observations and experiments. It was not until the time of Galileo and Newton that this was done. (v) We know, both from occasional references in his works and from remarks by other writers, that Archimedes was largely occupied in astronomical observations. He wrote a book, Περὶ Σφειροποιίας, on the construction of a celestial sphere, which is lost; and he constructed a sphere of the stars, and an orrery. These, after the capture of Syracuse, were taken by Marcellus to Rome, and were preserved as curiosities for at least two or three hundred years. This mere catalogue of his works will show how wonderful were his achieveC ments; but no one who has not actually read some of D his writings can form a just appreciation of his extraorA B dinary ability. This will be still further increased if we recollect that the only principles used by Archimedes, in addition to those contained in Euclid’s Elements and Conic sections, are that of all lines like ACB, ADB, . . . connecting two points A and B, the straight line is the shortest, and of the curved lines, the inner one ADB is shorter than the outer one ACB; together with two similar statements for space of three dimensions. In the old and medieval world Archimedes was reckoned as the first of mathematicians, but possibly the best tribute to his fame is the fact that those writers who have spoken most highly of his work and ability are those who have been themselves the most distinguished men of their own generation. Apollonius.1 The third great mathematician of this century 1

In addition to Zeuthen’s work and the other authorities mentioned in the footnote on p. 41, see Litterargeschichtliche Studien u ¨ber Euklid, by J. L. Heiberg, Leipzig, 1882. Editions of the extant works of Apollonius were issued by

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was Apollonius of Perga, who is chiefly celebrated for having produced a systematic treatise on the conic sections which not only included all that was previously known about them, but immensely extended the knowledge of these curves. This work was accepted at once as the standard text-book on the subject, and completely superseded the previous treatises of Menaechmus, Aristaeus, and Euclid which until that time had been in general use. We know very little of Apollonius himself. He was born about 260 b.c., and died about 200 b.c. He studied in Alexandria for many years, and probably lectured there; he is represented by Pappus as “vain, jealous of the reputation of others, and ready to seize every opportunity to depreciate them.” It is curious that while we know next to nothing of his life, or of that of his contemporary Eratosthenes, yet their nicknames, which were respectively epsilon and beta, have come down to us. Dr. Gow has ingeniously suggested that the lecture rooms at Alexandria were numbered, and that they always used the rooms numbered 5 and 2 respectively. Apollonius spent some years at Pergamum in Pamphylia, where a university had been recently established and endowed in imitation of that at Alexandria. There he met Eudemus and Attalus, to whom he subsequently sent each book of his conics as it came out with an explanatory note. He returned to Alexandria, and lived there till his death, which was nearly contemporaneous with that of Archimedes. In his great work on conic sections Apollonius so thoroughly investigated the properties of these curves that he left but little for his successors to add. But his proofs are long and involved, and I think most readers will be content to accept a short analysis of his work, and the assurance that his demonstrations are valid. Dr. Zeuthen believes that many of the properties enunciated were obtained in the first instance by the use of co-ordinate geometry, and that the demonstrations were translated subsequently into geometrical form. If this be so, we must suppose that the classical writers were familiar with some branches of analytical geometry—Dr. Zeuthen says the use of orthogonal and oblique co-ordinates, and of transformations depending on abridged notation—that this knowledge was confined to a limited school, and was finally lost. This is a mere conjecture and is unsupported by any direct evidence, but it has been accepted by some writers J. L. Heiberg in two volumes, Leipzig, 1890, 1893; and by E. Halley, Oxford, 1706 and 1710: an edition of the conics was published by T. L. Heath, Cambridge, 1896.

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as affording an explanation of the extent and arrangement of the work. The treatise contained about four hundred propositions, and was divided into eight books; we have the Greek text of the first four of these, and we also possess copies of the commentaries by Pappus and Eutocius on the whole work. In the ninth century an Arabic translation was made of the first seven books, which were the only ones then extant; we have two manuscripts of this version. The eighth book is lost. In the letter to Eudemus which accompanied the first book Apollonius says that he undertook the work at the request of Naucrates, a geometrician who had been staying with him at Alexandria, and, though he had given some of his friends a rough draft of it, he had preferred to revise it carefully before sending it to Pergamum. In the note which accompanied the next book, he asks Eudemus to read it and communicate it to others who can understand it, and in particular to Philonides, a certain geometrician whom the author had met at Ephesus. The first four books deal with the elements of the subject, and of these the first three are founded on Euclid’s previous work (which was itself based on the earlier treatises by Menaechmus and Aristaeus). Heracleides asserts that much of the matter in these books was stolen from an unpublished work of Archimedes, but a critical examination by Heiberg has shown that this is improbable.

P

A

M

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Apollonius begins by defining a cone on a circular base. He then investigates the different plane sections of it, and shows that they are divisible into three kinds of curves which he calls ellipses, parabolas, and hyperbolas. He proves the proposition that, if A, A0 be the vertices of a conic, and if P be any point on it, and P M the perpendicular drawn from P on AA0 , then (in the usual notation) the ratio M P 2 : AM . M A0 is constant in an ellipse or hyperbola, and the ratio M P 2 : AM is constant in a parabola. These are the characteristic properties on which almost all the rest of the work is based. He next shows that, if A be the vertex, l the latus rectum, and if AM and M P be the abscissa and ordinate of any point on a conic (see above figure), then M P 2 is less than, equal to, or greater than l . AM according as the conic is an ellipse, parabola, or hyperbola; hence the names which he gave to the curves and by which they are still known. He had no idea of the directrix, and was not aware that the parabola had a focus, but, with the exception of the propositions which involve these, his first three books contain most of the propositions which are found in modern text-books. In the fourth book he develops the theory of lines cut harmonically, and treats of the points of intersection of systems of conics. In the fifth book he commences with the theory of maxima and minima; applies it to find the centre of curvature at any point of a conic, and the evolute of the curve; and discusses the number of normals which can be drawn from a point to a conic. In the sixth book he treats of similar conics. The seventh and eighth books were given up to a discussion of conjugate diameters; the latter of these was conjecturally restored by E. Halley in his edition of 1710. The verbose explanations make the book repulsive to most modern readers; but the arrangement and reasoning are unexceptional, and it has been not unfitly described as the crown of Greek geometry. It is the work on which the reputation of Apollonius rests, and it earned for him the name of “the great geometrician.” Besides this immense treatise he wrote numerous shorter works; of course the books were written in Greek, but they are usually referred to by their Latin titles: those about which we now know anything are enumerated below. He was the author of a work on the problem “given two co-planar straight lines Aa and Bb, drawn through fixed points A and B; to draw a line Oab from a given point O outside them cutting them in a and b, so that Aa shall be to Bb in a given ratio.” He reduced the question to seventy-seven separate cases and gave an appropriate solution, with the aid of conics, for each case; this was published by

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E. Halley (translated from an Arabic copy) in 1706. He also wrote a treatise De Sectione Spatii (restored by E. Halley in 1706) on the same problem under the condition that the rectangle Aa . Bb was given. He wrote another entitled De Sectione Determinata (restored by R. Simson in 1749), dealing with problems such as to find a point P in a given straight line AB, so that P A2 shall be to P B in a given ratio. He wrote another De Tactionibus (restored by Vieta in 1600) on the construction of a circle which shall touch three given circles. Another work was his De Inclinationibus (restored by M. Ghetaldi in 1607) on the problem to draw a line so that the intercept between two given lines, or the circumferences of two given circles, shall be of a given length. He was also the author of a treatise in three books on plane loci, De Locis Planis (restored by Fermat in 1637, and by R. Simson in 1746), and of another on the regular solids. And, lastly, he wrote a treatise on unclassed incommensurables, being a commentary on the tenth book of Euclid. It is believed that in one or more of the lost books he used the method of conical projections. Besides these geometrical works he wrote on the methods of arithmetical calculation. All that we know of this is derived from some remarks of Pappus. Friedlein thinks that it was merely a sort of readyreckoner. It seems, however, more probable that Apollonius here suggested a system of numeration similar to that proposed by Archimedes, but proceeding by tetrads instead of octads, and described a notation for it. It will be noticed that our modern notation goes by hexads, a million = 106 , a billion = 1012 , a trillion = 1018 , etc. It is not impossible that Apollonius also pointed out that a decimal system of notation, involving only nine symbols, would facilitate numerical multiplications. Apollonius was interested in astronomy, and wrote a book on the stations and regressions of the planets of which Ptolemy made some use in writing the Almagest. He also wrote a treatise on the use and theory of the screw in statics. This is a long list, but I should suppose that most of these works were short tracts on special points. Like so many of his predecessors, he too gave a construction for finding two mean proportionals between two given lines, and thereby duplicating the cube. It was as follows. Let OA and OB be the given lines. Construct a rectangle OADB, of which they are adjacent sides. Bisect AB in C. Then, if with C as centre we can describe a circle cutting OA produced in a, and cutting OB produced in b, so that aDb shall be a straight line, the problem is effected. For it is easily shewn

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that Oa . Aa + CA2 = Ca2 . Ob . Bb + CB 2 = Cb2 . Oa . Aa = Ob . Bb. Oa : Ob = Bb : Aa.

Similarly Hence That is, b

D

B C O

A

a

But, by similar triangles,

Therefore

BD : Bb = Oa : Ob = Aa : AD. Oa : Bb = Bb : Aa = Aa : OB,

that is, Bb and Oa are the two mean proportionals between OA and OB. It is impossible to construct the circle whose centre is C by Euclidean geometry, but Apollonius gave a mechanical way of describing it. This construction is quoted by several Arabic writers. In one of the most brilliant passages of his Aper¸cu historique Chasles remarks that, while Archimedes and Apollonius were the most able geometricians of the old world, their works are distinguished by a contrast which runs through the whole subsequent history of geometry. Archimedes, in attacking the problem of the quadrature of curvilinear areas, established the principles of the geometry which rests on measurements; this naturally gave rise to the infinitesimal calculus, and in fact the method of exhaustions as used by Archimedes does not differ in principle from the method of limits as used by Newton. Apollonius, on the other hand, in investigating the properties of conic sections by

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means of transversals involving the ratio of rectilineal distances and of perspective, laid the foundations of the geometry of form and position. Eratosthenes.1 Among the contemporaries of Archimedes and Apollonius I may mention Eratosthenes. Born at Cyrene in 275 b.c., he was educated at Alexandria—perhaps at the same time as Archimedes, of whom he was a personal friend—and Athens, and was at an early age entrusted with the care of the university library at Alexandria, a post which probably he occupied till his death. He was the Admirable Crichton of his age, and distinguished for his athletic, literary, and scientific attainments: he was also something of a poet. He lost his sight by ophthalmia, then as now a curse of the valley of the Nile, and, refusing to live when he was no longer able to read, he committed suicide in 194 b.c. In science he was chiefly interested in astronomy and geodesy, and he constructed various astronomical instruments which were used for some centuries at the university. He suggested the calendar (now known as Julian), in which every fourth year contains 366 days; and he determined the obliquity of the ecliptic as 23◦ 510 2000 . He measured the length of a degree on the earth’s surface, making it to be about 79 miles, which is too long by nearly 10 miles, and thence calculated the circumference of the earth to be 252,000 stadia. If we take the Olympic stadium of 202 14 yards, this is equivalent to saying that the radius is about 4600 miles, but there was also an Egyptian stadium, and if he used this he estimated the radius as 3925 miles, which is very near the truth. The principle used in the determination is correct. Of Eratosthenes’s work in mathematics we have two extant illustrations: one in a description of an instrument to duplicate a cube, and the other in a rule he gave for constructing a table of prime numbers. The former is given in many books. The latter, called the “sieve of Eratosthenes,” was as follows: write down all the numbers from 1 upwards; then every second number from 2 is a multiple of 2 and may be cancelled; every third number from 3 is a multiple of 3 and may be cancelled; every fifth number from 5 is a multiple of 5 and may be cancelled; and so on. It has been estimated that it would involve working for about 300 hours to thus find the primes in the numbers from 1 to 1,000,000. The labour of determining whether any particular number 1

The works of Eratosthenes exist only in fragments. A collection of these was published by G. Bernhardy at Berlin in 1822: some additional fragments were printed by E. Hillier, Leipzig, 1872.

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is a prime may be, however, much shortened by observing that if a number can be expressed as the product of two factors, one must be less and the other greater than the square root of the number, unless the number is the square of a prime, in which case the two factors are equal. Hence every composite number must be divisible by a prime which is not greater than its square root. The second century before Christ. The third century before Christ, which opens with the career of Euclid and closes with the death of Apollonius, is the most brilliant era in the history of Greek mathematics. But the great mathematicians of that century were geometricians, and under their influence attention was directed almost solely to that branch of mathematics. With the methods they used, and to which their successors were by tradition confined, it was hardly possible to make any further great advance: to fill up a few details in a work that was completed in its essential parts was all that could be effected. It was not till after the lapse of nearly 1800 years that the genius of Descartes opened the way to any further progress in geometry, and I therefore pass over the numerous writers who followed Apollonius with but slight mention. Indeed it may be said roughly that during the next thousand years Pappus was the sole geometrician of great original ability; and during this long period almost the only other pure mathematicians of exceptional genius were Hipparchus and Ptolemy, who laid the foundations of trigonometry, and Diophantus, who laid those of algebra. Early in the second century, circ. 180 b.c., we find the names of three mathematicians—Hypsicles, Nicomedes, and Diocles—who in their own day were famous. Hypsicles. The first of these was Hypsicles, who added a fourteenth book to Euclid’s Elements in which the regular solids were discussed. In another small work, entitled Risings, we find for the first time in Greek mathematics a right angle divided in the Babylonian manner into ninety degrees; possibly Eratosthenes may have previously estimated angles by the number of degrees they contain, but this is only a matter of conjecture. Nicomedes. The second was Nicomedes, who invented the curve known as the conchoid or the shell-shaped curve. If from a fixed point S a line be drawn cutting a given fixed straight line in Q, and if P be taken on SQ so that the length QP is constant (say d), then the locus of

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P is the conchoid. Its equation may be put in the form r = a sec θ ± d. It is easy with its aid to trisect a given angle or to duplicate a cube; and this no doubt was the cause of its invention. Diocles. The third of these mathematicians was Diocles, the inventor of the curve known as the cissoid or the ivy-shaped curve, which, like the conchoid, was used to give a solution of the duplication problem. He defined it thus: let AOA0 and BOB 0 be two fixed diameters of a circle at right angles to one another. Draw two chords QQ0 and RR0 parallel to BOB 0 and equidistant from it. Then the locus of the intersection of AR and QQ0 will be the cissoid. Its equation can be expressed in the form y 2 (2a − x) = x3 . The curve may be used to duplicate the cube. For, if OA and OE be the two lines between which it is required to insert two geometrical means, and if, in the figure constructed as above, A0 E cut the cissoid in P , and AP cut OB in D, we have OD3 = OA2 . OE. Thus OD is one of the means required, and the other mean can be found at once. Diocles also solved (by the aid of conic sections) a problem which had been proposed by Archimedes, namely, to draw a plane which will divide a sphere into two parts whose volumes shall bear to one another a given ratio. Perseus. Zenodorus. About a quarter of a century later, say about 150 b.c., Perseus investigated the various plane sections of the anchor-ring, and Zenodorus wrote a treatise on isoperimetrical figures. Part of the latter work has been preserved; one proposition which will serve to show the nature of the problems discussed is that “of segments of circles, having equal arcs, the semicircle is the greatest.” Towards the close of this century we find two mathematicians who, by turning their attention to new subjects, gave a fresh stimulus to the study of mathematics. These were Hipparchus and Hero. Hipparchus.1 Hipparchus was the most eminent of Greek astronomers—his chief predecessors being Eudoxus, Aristarchus, Archimedes, and Eratosthenes. Hipparchus is said to have been born about 160 b.c. at Nicaea in Bithynia; it is probable that he spent some years at Alexandria, but finally he took up his abode at Rhodes where he made most of his observations. Delambre has obtained an ingenious 1

See C. Manitius, Hipparchi in Arati et Eudoxi Phaenomena Commentarii, Leipzig, 1894, and J. B. J. Delambre, Histoire de l’astronomie ancienne, Paris, 1817, vol. i, pp. 106–189. S. P. Tannery in his Recherches sur l’histoire de l’astronomie ancienne, Paris, 1893, argues that the work of Hipparchus has been overrated, but I have adopted the view of the majority of writers on the subject.

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confirmation of the tradition which asserted that Hipparchus lived in the second century before Christ. Hipparchus in one place says that the longitude of a certain star η Canis observed by him was exactly 90◦ , and it should be noted that he was an extremely careful observer. Now in 1750 it was 116◦ 40 1000 , and, as the first point of Aries regredes at the rate of 50.200 a year, the observation was made about 120 b.c. Except for a short commentary on a poem of Aratus dealing with astronomy all his works are lost, but Ptolemy’s great treatise, the Almagest, described below, was founded on the observations and writings of Hipparchus, and from the notes there given we infer that the chief discoveries of Hipparchus were as follows. He determined the duration of the year to within six minutes of its true value. He calculated the inclination of the ecliptic and equator as 23◦ 510 ; it was actually at that time 23◦ 460 . He estimated the annual precession of the equinoxes as 5900 ; it is 50.200 . He stated the lunar parallax as 570 , which is nearly correct. He worked out the eccentricity of the solar orbit as 1/24; it is very approximately 1/30. He determined the perigee and mean motion of the sun and of the moon, and he calculated the extent of the shifting of the plane of the moon’s motion. Finally he obtained the synodic periods of the five planets then known. I leave the details of his observations and calculations to writers who deal specially with astronomy such as Delambre; but it may be fairly said that this work placed the subject for the first time on a scientific basis. To account for the lunar motion Hipparchus supposed the moon to move with uniform velocity in a circle, the earth occupying a position near (but not at) the centre of this circle. This is equivalent to saying that the orbit is an epicycle of the first order. The longitude of the moon obtained on this hypothesis is correct to the first order of small quantities for a few revolutions. To make it correct for any length of time Hipparchus further supposed that the apse line moved forward about 3◦ a month, thus giving a correction for eviction. He explained the motion of the sun in a similar manner. This theory accounted for all the facts which could be determined with the instruments then in use, and in particular enabled him to calculate the details of eclipses with considerable accuracy. He commenced a series of planetary observations to enable his successors to frame a theory to account for their motions; and with great perspicacity he predicted that to do this it would be necessary to introduce epicycles of a higher order, that is, to introduce three or more circles the centre of each successive one moving uniformly along the

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circumference of the preceding one. He also formed a list of 1080 of the fixed stars. It is said that the sudden appearance in the heavens of a new and brilliant star called his attention to the need of such a catalogue; and the appearance of such a star during his lifetime is confirmed by Chinese records. No further advance in the theory of astronomy was made until the time of Copernicus, though the principles laid down by Hipparchus were extended and worked out in detail by Ptolemy. Investigations such as these naturally led to trigonometry, and Hipparchus must be credited with the invention of that subject. It is known that in plane trigonometry he constructed a table of chords of arcs, which is practically the same as one of natural sines; and that in spherical trigonometry he had some method of solving triangles: but his works are lost, and we can give no details. It is believed, however, that the elegant theorem, printed as Euc. vi, d, and generally known as Ptolemy’s Theorem, is due to Hipparchus and was copied from him by Ptolemy. It contains implicitly the addition formulae for sin(A ± B) and cos(A ± B); and Carnot showed how the whole of elementary plane trigonometry could be deduced from it. I ought also to add that Hipparchus was the first to indicate the position of a place on the earth by means of its latitude and longitude. Hero.1 The second of these mathematicians was Hero of Alexandria, who placed engineering and land-surveying on a scientific basis. He was a pupil of Ctesibus, who invented several ingenious machines, and is alluded to as if he were a mathematician of note. It is not likely that Hero flourished before 80 b.c., but the precise period at which he lived is uncertain. In pure mathematics Hero’s principal and most characteristic work consists of (i) some elementary geometry, with applications to the determination of the areas of fields of given shapes; (ii) propositions on finding the volumes of certain solids, with applications to theatres, 1

See Recherches sur la vie et les ouvrages d’H´eron d’Alexandrie by T. H. Martin in vol. iv of M´emoires pr´esent´es . . . ` a l’acad´emie d’inscriptions, Paris, 1854; see also Loria, book iii, chap. v, pp. 107–128, and Cantor, chaps. xviii, xix. On the work entitled Definitions, which is attributed to Hero, see S. P. Tannery, chaps. xiii, xiv, and an article by G. Friedlein in Boncompagni’s Bulletino di bibliografia March 1871, vol. iv, pp. 93–126. Editions of the extant works of Hero were published in Teubner’s series, Leipzig, 1899, 1900, 1903. An English translation of the Πνευματικά was published by B. Woodcroft and J. G. Greenwood, London, 1851: drawings of the apparatus are inserted.

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baths, banquet-halls, and so on; (iii) a rule to find the height of an inaccessible object; and (iv) tables of weights and measures. He invented a solution of the duplication problem which is practically the same as that which Apollonius had already discovered. Some commentators think that he knew how to solve a quadratic equation even when the coefficients were not numerical; but this is doubtful. He proved the formula that the area of a triangle is equal to {s(s − a)(s − b)(s − c)}1/2 , where s is the semiperimeter, and a, b, c, the lengths of the sides, and gave as an illustration a triangle whose sides were in the ratio 13:14:15. He seems to have been acquainted with the trigonometry of Hipparchus, and the values of cot 2π/n are computed for various values of n, but he nowhere quotes a formula or expressly uses the value of the sine; it is probable that like the later Greeks he regarded trigonometry as forming an introduction to, and being an integral part of, astronomy. A

E

F O

B

D

L

C

H

K

The following is the manner in which he solved1 the problem to 1

In his Dioptra, Hultsch, part viii, pp. 235–237. It should be stated that some

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find the area of a triangle ABC the length of whose sides are a, b, c. Let s be the semiperimeter of the triangle. Let the inscribed circle touch the sides in D, E, F , and let O be its centre. On BC produced take H so that CH = AF , therefore BH = s. Draw OK at right angles to OB, and CK at right angles to BC; let them meet in K. The area ABC or 4 is equal to the sum of the areas OBC, OCA, OAB = 12 ar + 21 br + 12 cr = sr, that is, is equal to BH . OD. He then shews that the angle OAF = angle CBK; hence the triangles OAF and CBK are similar. ∴ BC : CK = AF : OF = CH : OD, ∴ BC : CH = CK : OD = CL : LD, ∴ BH : CH = CD : LD, ∴ BH 2 : CH . BH = CD . BD : LD . BD = CD . BD : OD2 . Hence 1

1

4 = BH . OD = {CH . BH . CD . BD} 2 = {(s − a)s(s − c)(s − b)} 2 . In applied mathematics Hero discussed the centre of gravity, the five simple machines, and the problem of moving a given weight with a given power; and in one place he suggested a way in which the power of a catapult could be tripled. He also wrote on the theory of hydraulic machines. He described a theodolite and cyclometer, and pointed out various problems in surveying for which they would be useful. But the most interesting of his smaller works are his Πνευματικά and Αὐτόματα, containing descriptions of about 100 small machines and mechanical toys, many of which are ingenious. In the former there is an account of a small stationary steam-engine which is of the form now known as Avery’s patent: it was in common use in Scotland at the beginning of this century, but is not so economical as the form introduced by Watt. There is also an account of a double forcing pump to be used as a fire-engine. It is probable that in the hands of Hero these instruments never got beyond models. It is only recently that general attention has been directed to his discoveries, though Arago had alluded to them in his ´eloge on Watt. All this is very different from the classical geometry and arithmetic of Euclid, or the mechanics of Archimedes. Hero did nothing to extend a knowledge of abstract mathematics; he learnt all that the text-books critics think that this is an interpolation, and is not due to Hero.

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of the day could teach him, but he was interested in science only on account of its practical applications, and so long as his results were true he cared nothing for the logical accuracy of the process by which he arrived at them. Thus, in finding the area of a triangle, he took the square root of the product of four lines. The classical Greek geometricians permitted the use of the square and the cube of a line because these could be represented geometrically, but a figure of four dimensions is inconceivable, and certainly they would have rejected a proof which involved such a conception. The first century before Christ. The successors of Hipparchus and Hero did not avail themselves of the opportunity thus opened of investigating new subjects, but fell back on the well-worn subject of geometry. Amongst the more eminent of these later geometricians were Theodosius and Dionysodorus, both of whom flourished about 50 b.c. Theodosius. Theodosius was the author of a complete treatise on the geometry of the sphere, and of two works on astronomy.1 Dionysodorus. Dionysodorus is known to us only by his solution2 of the problem to divide a hemisphere by a plane parallel to its base into two parts, whose volumes shall be in a given ratio. Like the solution by Diocles of the similar problem for a sphere above alluded to, it was effected by the aid of conic sections. Pliny says that Dionysodorus determined the length of the radius of the earth approximately as 42,000 stadia, which, if we take the Olympic stadium of 202 14 yards, is a little less than 5000 miles; we do not know how it was obtained. This may be compared with the result given by Eratosthenes and mentioned above. End of the First Alexandrian School. The administration of Egypt was definitely undertaken by Rome in 30 b.c. The closing years of the dynasty of the Ptolemies and the earlier years of the Roman occupation of the country were marked by much disorder, civil and political. The studies of the university were 1

The work on the sphere was edited by I. Barrow, Cambridge, 1675, and by E. Nizze, Berlin, 1852. The works on astronomy were published by Dasypodius in 1572. 2 It is reproduced in H. Suter’s Geschichte der mathematischen Wissenschaften, second edition, Z¨ urich, 1873, p. 101.

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naturally interrupted, and it is customary to take this time as the close of the first Alexandrian school.

78

CHAPTER V. the second alexandrian school.1 30 b.c.–641 a.d. I concluded the last chapter by stating that the first school of Alexandria may be said to have come to an end at about the same time as the country lost its nominal independence. But, although the schools at Alexandria suffered from the disturbances which affected the whole Roman world in the transition, in fact if not in name, from a republic to an empire, there was no break of continuity; the teaching in the university was never abandoned; and as soon as order was again established, students began once more to flock to Alexandria. This time of confusion was, however, contemporaneous with a change in the prevalent views of philosophy which thenceforward were mostly neo-platonic or neo-pythagorean, and it therefore fitly marks the commencement of a new period. These mystical opinions reacted on the mathematical school, and this may partially account for the paucity of good work. Though Greek influence was still predominant and the Greek language always used, Alexandria now became the intellectual centre for most of the Mediterranean nations which were subject to Rome. It should be added, however, that the direct connection with it of many of the mathematicians of this time is at least doubtful, but their knowledge was ultimately obtained from the Alexandrian teachers, and they are usually described as of the second Alexandrian school. Such mathematics as were taught at Rome were derived from Greek sources, and we may therefore conveniently consider their extent in connection with this chapter. 1

For authorities, see footnote above on p. 41. All dates given hereafter are to be taken as anno domini unless the contrary is expressly stated.

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The first century after Christ. There is no doubt that throughout the first century after Christ geometry continued to be that subject in science to which most attention was devoted. But by this time it was evident that the geometry of Archimedes and Apollonius was not capable of much further extension; and such geometrical treatises as were produced consisted mostly of commentaries on the writings of the great mathematicians of a preceding age. In this century the only original works of any ability of which we know anything were two by Serenus and one by Menelaus. Serenus. Menelaus. Those by Serenus of Antissa or of Antinoe, circ. 70, are on the plane sections of the cone and cylinder,1 in the course of which he lays down the fundamental proposition of transversals. That by Menelaus of Alexandria, circ. 98, is on spherical trigonometry, investigated in the Euclidean method.2 The fundamental theorem on which the subject is based is the relation between the six segments of the sides of a spherical triangle, formed by the arc of a great circle which cuts them [book iii, prop. 1]. Menelaus also wrote on the calculation of chords, that is, on plane trigonometry; this is lost. Nicomachus. Towards the close of this century, circ. 100, a Jew, Nicomachus, of Gerasa, published an Arithmetic,3 which (or rather the Latin translation of it) remained for a thousand years a standard authority on the subject. Geometrical demonstrations are here abandoned, and the work is a mere classification of the results then known, with numerical illustrations: the evidence for the truth of the propositions enunciated, for I cannot call them proofs, being in general an induction from numerical instances. The object of the book is the study of the properties of numbers, and particularly of their ratios. Nicomachus commences with the usual distinctions between even, odd, prime, and perfect numbers; he next discusses fractions in a somewhat clumsy manner; he then turns to polygonal and to solid numbers; and finally treats of ratio, proportion, and the progressions. Arithmetic of this kind is usually termed Boethian, and the work of Boethius on it was a recognised text-book in the middle ages. 1

These have been edited by J. L. Heiberg, Leipzig, 1896; and by E. Halley, Oxford, 1710. 2 This was translated by E. Halley, Oxford, 1758. 3 The work has been edited by R. Hoche, Leipzig, 1866.

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The second century after Christ. Theon. Another text-book on arithmetic on much the same lines as that of Nicomachus was produced by Theon of Smyrna, circ. 130. It formed the first book of his work1 on mathematics, written with the view of facilitating the study of Plato’s writings. Thymaridas. Another mathematician, reckoned by some writers as of about the same date as Theon, was Thymaridas, who is worthy of notice from the fact that he is the earliest known writer who explicitly enunciates an algebraical theorem. He states that, if the sum of any number of quantities be given, and also the sum of every pair which contains one of them, then this quantity is equal to one (n − 2)th part of the difference between the sum of these pairs and the first given sum. Thus, if

and if then

x1 + x2 + . . . + xn = S, x1 + x2 = s2 , x1 + x3 = s3 , . . . , and x1 + xn = sn , x1 = (s2 + s3 + . . . + sn − S)/(n − 2).

He does not seem to have used a symbol to denote the unknown quantity, but he always represents it by the same word, which is an approximation to symbolism. Ptolemy.2 About the same time as these writers Ptolemy of Alexandria, who died in 168, produced his great work on astronomy, which will preserve his name as long as the history of science endures. This treatise is usually known as the Almagest: the name is derived from the Arabic title al midschisti, which is said to be a corruption of μεγίστη [μαθηματική] σύνταξις. The work is founded on the writings of Hipparchus, and, though it did not sensibly advance the theory of the subject, it presents the views of the older writer with a completeness and elegance which will always make it a standard treatise. We gather from it that Ptolemy made observations at Alexandria from the years 1

The Greek text of those parts which are now extant, with a French translation, was issued by J. Dupuis, Paris, 1892. 2 See the article Ptolemaeus Claudius, by A. De Morgan in Smith’s Dictionary of Greek and Roman Biography, London, 1849; S. P. Tannery, Recherches sur l’histoire de l’astronomie ancienne, Paris, 1893; and J. B. J. Delambre, Histoire de l’astronomie ancienne, Paris, 1817, vol. ii. An edition of all the works of Ptolemy which are now extant was published at Bˆ ale in 1551. The Almagest with various minor works was edited by M. Halma, 12 vols. Paris, 1813–28, and a new edition, in two volumes, by J. L. Heiberg, Leipzig, 1898, 1903, 1907.

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125 to 150; he, however, was but an indifferent practical astronomer, and the observations of Hipparchus are generally more accurate than those of his expounder. The work is divided into thirteen books. In the first book Ptolemy discusses various preliminary matters; treats of trigonometry, plane or spherical; gives a table of chords, that is, of natural sines (which is substantially correct and is probably taken from the lost work of Hipparchus); and explains the obliquity of the ecliptic; in this book he uses degrees, minutes, and seconds as measures of angles. The second book is devoted chiefly to phenomena depending on the spherical form of the earth: he remarks that the explanations would be much simplified if the earth were supposed to rotate on its axis once a day, but states that this hypothesis is inconsistent with known facts. In the third book he explains the motion of the sun round the earth by means of excentrics and epicycles: and in the fourth and fifth books he treats the motion of the moon in a similar way. The sixth book is devoted 17 , as to the theory of eclipses; and in it he gives 3◦ 80 3000 , that is 3 120 the approximate value of π, which is equivalent to taking it equal to 3.1416. The seventh and eighth books contain a catalogue (probably copied from Hipparchus) of 1028 fixed stars determined by indicating those, three or more, that appear to be in a plane passing through the observer’s eye: and in another work Ptolemy added a list of annual sidereal phenomena. The remaining books are given up to the theory of the planets. This work is a splendid testimony to the ability of its author. It became at once the standard authority on astronomy, and remained so till Copernicus and Kepler shewed that the sun and not the earth must be regarded as the centre of the solar system. The idea of excentrics and epicycles on which the theories of Hipparchus and Ptolemy are based has been often ridiculed in modern times. No doubt at a later time, when more accurate observations had been made, the necessity of introducing epicycle on epicycle in order to bring the theory into accordance with the facts made it very complicated. But De Morgan has acutely observed that in so far as the ancient astronomers supposed that it was necessary to resolve every celestial motion into a series of uniform circular motions they erred greatly, but that, if the hypothesis be regarded as a convenient way of expressing known facts, it is not only legitimate but convenient. The theory suffices to describe either the angular motion of the heavenly bodies or their change in distance. The ancient astronomers were concerned only

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with the former question, and it fairly met their needs; for the latter question it is less convenient. In fact it was as good a theory as for their purposes and with their instruments and knowledge it was possible to frame, and corresponds to the expression of a given function as a sum of sines or cosines, a method which is of frequent use in modern analysis. In spite of the trouble taken by Delambre it is almost impossible to separate the results due to Hipparchus from those due to Ptolemy. But Delambre and De Morgan agree in thinking that the observations quoted, the fundamental ideas, and the explanation of the apparent solar motion are due to Hipparchus; while all the detailed explanations and calculations of the lunar and planetary motions are due to Ptolemy. E A

F

B

N

M C

G

D

H The Almagest shews that Ptolemy was a geometrician of the first rank, though it is with the application of geometry to astronomy that he is chiefly concerned. He was also the author of numerous other treatises. Amongst these is one on pure geometry in which he proposed to cancel Euclid’s postulate on parallel lines, and to prove it in the following manner. Let the straight line EF GH meet the two straight lines AB and CD so as to make the sum of the angles BF G and F GD equal to two right angles. It is required to prove that AB and CD are parallel. If possible let them not be parallel, then they will meet when produced say at M (or N ). But the angle AF G is the supplement of BF G, and is therefore equal to F GD: similarly the angle F GC is equal to the angle BF G. Hence the sum of the angles AF G and F GC is equal to two right angles, and the lines BA and DC will therefore if produced meet at N (or M ). But two straight lines cannot enclose a space, therefore AB and CD cannot meet when produced, that is, they are parallel. Conversely, if AB and CD be parallel, then AF and CG are not less parallel than F B and GD; and therefore whatever be the sum of the angles AF G and F GC such also must be the sum of

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the angles F GD and BF G. But the sum of the four angles is equal to four right angles, and therefore the sum of the angles BF G and F GD must be equal to two right angles. Ptolemy wrote another work to shew that there could not be more than three dimensions in space: he also discussed orthographic and stereographic projections with special reference to the construction of sun-dials. He wrote on geography, and stated that the length of one degree of latitude is 500 stadia. A book on sound is sometimes attributed to him, but on doubtful authority. The third century after Christ. Pappus. Ptolemy had shewn not only that geometry could be applied to astronomy, but had indicated how new methods of analysis like trigonometry might be thence developed. He found however no successors to take up the work he had commenced so brilliantly, and we must look forward 150 years before we find another geometrician of any eminence. That geometrician was Pappus who lived and taught at Alexandria about the end of the third century. We know that he had numerous pupils, and it is probable that he temporarily revived an interest in the study of geometry. Pappus wrote several books, but the only one which has come down to us is his Συναγωγή,1 a collection of mathematical papers arranged in eight books of which the first and part of the second have been lost. This collection was intended to be a synopsis of Greek mathematics together with comments and additional propositions by the editor. A careful comparison of various extant works with the account given of them in this book shews that it is trustworthy, and we rely largely on it for our knowledge of other works now lost. It is not arranged chronologically, but all the treatises on the same subject are grouped together, and it is most likely that it gives roughly the order in which the classical authors were read at Alexandria. Probably the first book, which is now lost, was on arithmetic. The next four books deal with geometry exclusive of conic sections; the sixth with astronomy including, as subsidiary subjects, optics and trigonometry; the seventh with analysis, conics, and porisms; and the eighth with mechanics. The last two books contain a good deal of original work by Pappus; at the same time it should be remarked that in two or three cases he 1

It has been published by F. Hultsch, Berlin, 1876–8.

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has been detected in appropriating proofs from earlier authors, and it is possible he may have done this in other cases. Subject to this suspicion we may say that Pappus’s best work is in geometry. He discovered the directrix in the conic sections, but he investigated only a few isolated properties: the earliest comprehensive account was given by Newton and Boscovich. As an illustration of his power I may mention that he solved [book vii, prop. 107] the problem to inscribe in a given circle a triangle whose sides produced shall pass through three collinear points. This question was in the eighteenth century generalised by Cramer by supposing the three given points to be anywhere; and was considered a difficult problem.1 It was sent in 1742 as a challenge to Castillon, and in 1776 he published a solution. Lagrange, Euler, Lhulier, Fuss, and Lexell also gave solutions in 1780. A few years later the problem was set to a Neapolitan lad A. Giordano, who was only 16 but who had shewn marked mathematical ability, and he extended it to the case of a polygon of n sides which pass through n given points, and gave a solution both simple and elegant. Poncelet extended it to conics of any species and subject to other restrictions. In mechanics Pappus shewed that the centre of mass of a triangular lamina is the same as that of an inscribed triangular lamina whose vertices divide each of the sides of the original triangle in the same ratio. He also discovered the two theorems on the surface and volume of a solid of revolution which are still quoted in text-books under his name: these are that the volume generated by the revolution of a curve about an axis is equal to the product of the area of the curve and the length of the path described by its centre of mass; and the surface is equal to the product of the perimeter of the curve and the length of the path described by its centre of mass. The problems above mentioned are but samples of many brilliant but isolated theorems which were enunciated by Pappus. His work as a whole and his comments shew that he was a geometrician of power; but it was his misfortune to live at a time when but little interest was taken in geometry, and when the subject, as then treated, had been practically exhausted. Possibly a small tract2 on multiplication and division of sexagesimal 1

For references to this problem see a note by H. Brocard in L’Interm´ediaire des math´ematiciens, Paris, 1904, vol. xi, pp. 219–220. 2 It was edited by C. Henry, Halle, 1879, and is valuable as an illustration of practical Greek arithmetic.

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fractions, which would seem to have been written about this time, is due to Pappus. The fourth century after Christ. Throughout the second and third centuries, that is, from the time of Nicomachus, interest in geometry had steadily decreased, and more and more attention had been paid to the theory of numbers, though the results were in no way commensurate with the time devoted to the subject. It will be remembered that Euclid used lines as symbols for any magnitudes, and investigated a number of theorems about numbers in a strictly scientific manner, but he confined himself to cases where a geometrical representation was possible. There are indications in the works of Archimedes that he was prepared to carry the subject much further: he introduced numbers into his geometrical discussions and divided lines by lines, but he was fully occupied by other researches and had no time to devote to arithmetic. Hero abandoned the geometrical representation of numbers, but he, Nicomachus, and other later writers on arithmetic did not succeed in creating any other symbolism for numbers in general, and thus when they enunciated a theorem they were content to verify it by a large number of numerical examples. They doubtless knew how to solve a quadratic equation with numerical coefficients—for, as pointed out above, geometrical solutions of the equations ax2 − bx + c = 0 and ax2 + bx − c = 0 are given in Euc. vi, 28 and 29—but probably this represented their highest attainment. It would seem then that, in spite of the time given to their study, arithmetic and algebra had not made any sensible advance since the time of Archimedes. The problems of this kind which excited most interest in the third century may be illustrated from a collection of questions, printed in the Palatine Anthology, which was made by Metrodorus at the beginning of the next century, about 310. Some of them are due to the editor, but some are of an anterior date, and they fairly illustrate the way in which arithmetic was leading up to algebraical methods. The following are typical examples. “Four pipes discharge into a cistern: one fills it in one day; another in two days; the third in three days; the fourth in four days: if all run together how soon will they fill the cistern?” “Demochares has lived a fourth of his life as a boy; a fifth as a youth; a third as a man; and has spent thirteen years in his dotage: how old is he?” “Make a crown of gold, copper, tin, and iron weighing 60 minae: gold and copper shall be two-thirds of it; gold

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and tin three-fourths of it; and gold and iron three-fifths of it: find the weights of the gold, copper, tin, and iron which are required.” The last is a numerical illustration of Thymaridas’s theorem quoted above. It is believed that these problems were solved by rhetorical algebra, that is, by a process of algebraical reasoning expressed in words and without the use of any symbols. This, according to Nesselmann, is the first stage in the development of algebra, and we find it used both by Ahmes and by the earliest Arabian, Persian, and Italian algebraists: examples of its use in the solution of a geometrical problem and in the rule for the solution of a quadratic equation are given later.1 On this view then a rhetorical algebra had been gradually evolved by the Greeks, or was then in process of evolution. Its development was however very imperfect. Hankel, who is no unfriendly critic, says that the results attained as the net outcome of the work of six centuries on the theory of numbers are, whether we look at the form or the substance, unimportant or even childish, and are not in any way the commencement of a science. In the midst of this decaying interest in geometry and these feeble attempts at algebraic arithmetic, a single algebraist of marked originality suddenly appeared who created what was practically a new science. This was Diophantus who introduced a system of abbreviations for those operations and quantities which constantly recur, though in using them he observed all the rules of grammatical syntax. The resulting science is called by Nesselmann syncopated algebra: it is a sort of shorthand. Broadly speaking, it may be said that European algebra did not advance beyond this stage until the close of the sixteenth century. Modern algebra has progressed one stage further and is entirely symbolic; that is, it has a language of its own and a system of notation which has no obvious connection with the things represented, while the operations are performed according to certain rules which are distinct from the laws of grammatical construction. Diophantus.2 All that we know of Diophantus is that he lived at Alexandria, and that most likely he was not a Greek. Even the date of his career is uncertain; it cannot reasonably be put before the middle of the third century, and it seems probable that he was alive in the early 1

See below, pp. 168, 174. A critical edition of the collected works of Diophantus was edited by S. P. Tannery, 2 vols., Leipzig, 1893; see also Diophantos of Alexandria, by T. L. Heath, Cambridge, 1885; and Loria, book v, chap. v, pp. 95–158. 2

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years of the fourth century, that is, shortly after the death of Pappus. He was 84 when he died. In the above sketch of the lines on which algebra has developed I credited Diophantus with the invention of syncopated algebra. This is a point on which opinions differ, and some writers believe that he only systematized the knowledge which was familiar to his contemporaries. In support of this latter opinion it may be stated that Cantor thinks that there are traces of the use of algebraic symbolism in Pappus, and are used Freidlein mentions a Greek papyrus in which the signs / and for addition and subtraction respectively; but no other direct evidence for the non-originality of Diophantus has been produced, and no ancient author gives any sanction to this opinion. Diophantus wrote a short essay on polygonal numbers; a treatise on algebra which has come down to us in a mutilated condition; and a work on porisms which is lost. The Polygonal Numbers contains ten propositions, and was probably his earliest work. In this he reverts to the classical system by which numbers are represented by lines, a construction is (if necessary) made, and a strictly deductive proof follows: it may be noticed that in it he quotes propositions, such as Euc. ii, 3, and ii, 8, as referring to numbers and not to magnitudes. His chief work is his Arithmetic. This is really a treatise on algebra; algebraic symbols are used, and the problems are treated analytically. Diophantus tacitly assumes, as is done in nearly all modern algebra, that the steps are reversible. He applies this algebra to find solutions (though frequently only particular ones) of several problems involving numbers. I propose to consider successively the notation, the methods of analysis employed, and the subject-matter of this work. First, as to the notation. Diophantus always employed a symbol to represent the unknown quantity in his equations, but as he had only one symbol he could not use more than one unknown at a time.1 The unknown quantity is called ὁ ἀριθμός, and is represented by 0 or o0 . It is usually printed as ς. In the plural it is denoted by ςς or ςς o`ι . This symbol may be a corruption of αρ , or perhaps it may be the final sigma of this word, or possibly it may stand for the word σωρός a heap.2 The square of the unknown is called δύναμις, and denoted by δ υ¯ : the cube 1

See, however, below, page 90, example (iii), for an instance of how he treated a problem involving two unknown quantities. 2 See above, page 4.

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κύβος, and denoted by κυ ; and so on up to the sixth power. The coefficients of the unknown quantity and its powers are numbers, and a numerical coefficient is written immediately after the quantity it multiplies: thus ς 0 α ¯ = x, and ςς oι ια = ςς ια = 11x. An absolute term is regarded as a certain number of units or μονάδες which are represented by µoˆ: thus µoˆα ¯ = 1, µoˆια = 11. There is no sign for addition beyond juxtaposition. Subtraction is represented by , and this symbol affects all the symbols that follow it. Equality is represented by ι. Thus ψ

κυˆ α ¯ ςς η¯ δ oˆ¯ µoˆα ¯ ι ςα ¯ (x3 + 8x) − (5x2 + 1) = x. ψ

represents

Diophantus also introduced a somewhat similar notation for fractions involving the unknown quantity, but into the details of this I need not here enter. It will be noticed that all these symbols are mere abbreviations for words, and Diophantus reasons out his proofs, writing these abbreviations in the middle of his text. In most manuscripts there is a marginal summary in which the symbols alone are used and which is really symbolic algebra; but probably this is the addition of some scribe of later times. This introduction of a contraction or a symbol instead of a word to represent an unknown quantity marks a greater advance than anyone not acquainted with the subject would imagine, and those who have never had the aid of some such abbreviated symbolism find it almost impossible to understand complicated algebraical processes. It is likely enough that it might have been introduced earlier, but for the unlucky system of numeration adopted by the Greeks by which they used all the letters of the alphabet to denote particular numbers and thus made it impossible to employ them to represent any number. Next, as to the knowledge of algebraic methods shewn in the book. Diophantus commences with some definitions which include an explanation of his notation, and in giving the symbol for minus he states that a subtraction multiplied by a subtraction gives an addition; by this he means that the product of −b and −d in the expansion of (a − b)(c − d) is +bd, but in applying the rule he always takes care that the numbers a, b, c, d are so chosen that a is greater than b and c is greater than d. The whole of the work itself, or at least as much as is now extant, is devoted to solving problems which lead to equations. It contains

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rules for solving a simple equation of the first degree and a binomial quadratic. Probably the rule for solving any quadratic equation was given in that part of the work which is now lost, but where the equation is of the form ax2 +bx+c = 0 he seems to have multiplied by a and then “completed the square” in much the same way as is now done: when the roots are negative or irrational the equation is rejected as “impossible,” and even when both roots are positive he never gives more than one, always taking the positive value of the square root. Diophantus solves one cubic equation, namely, x3 + x = 4x2 + 4 [book vi, prob. 19]. The greater part of the work is however given up to indeterminate equations between two or three variables. When the equation is between two variables, then, if it be of the first degree, he assumes a suitable value for one variable and solves the equation for the other. Most of his equations are of the form y 2 = Ax2 + Bx + C. Whenever A or C is equal to zero, he is able to solve the equation completely. When this is not the case, then, if A = a2 , he assumes y = ax + m; if C = c2 , he assumes y = mx + c; and lastly, if the equation can be put in the form y 2 = (ax ± b)2 + c2 , he assumes y = mx: where in each case m has some particular numerical value suitable to the problem under consideration. A few particular equations of a higher order occur, but in these he generally alters the problem so as to enable him to reduce the equation to one of the above forms. The simultaneous indeterminate equations involving three variables, or “double equations” as he calls them, which he considers are of the forms y 2 = Ax2 + Bx + C and z 2 = ax2 + bx + c. If A and a both vanish, he solves the equations in one of two ways. It will be enough to give one of his methods which is as follows: he subtracts and thus gets an equation of the form y 2 − z 2 = mx + n; hence, if y ± z = λ, then y ∓ z = (mx + n)/λ; and solving he finds y and z. His treatment of “double equations” of a higher order lacks generality and depends on the particular numerical conditions of the problem. Lastly, as to the matter of the book. The problems he attacks and the analysis he uses are so various that they cannot be described concisely and I have therefore selected five typical problems to illustrate his methods. What seems to strike his critics most is the ingenuity with which he selects as his unknown some quantity which leads to equations such as he can solve, and the artifices by which he finds numerical solutions of his equations. I select the following as characteristic examples. (i) Find four numbers, the sum of every arrangement three at a time

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being given; say 22, 24, 27, and 20 [book i, prob. 17]. Let x be the sum of all four numbers; hence the numbers are x − 22, x − 24, x − 27, and x − 20. ∴ x = (x − 22) + (x − 24) + (x − 27) + (x − 20). ∴ x = 31. ∴ the numbers are 9, 7, 4, and 11. (ii) Divide a number, such as 13 which is the sum of two squares 4 and 9, into two other squares [book ii, prob. 10]. He says that since the given squares are 22 and 32 he will take (x + 2)2 and (mx − 3)2 as the required squares, and will assume m = 2. ∴ (x + 2)2 + (2x − 3)2 = 13. ∴ x = 8/5. ∴ the required squares are 324/25 and 1/25. (iii) Find two squares such that the sum of the product and either is a square [book ii, prob. 29]. Let x2 and y 2 be the numbers. Then x2 y 2 + y 2 and x2 y 2 + x2 are squares. The first will be a square if x2 +1 be a square, which he assumes may be taken equal to (x − 2)2 , hence x = 3/4. He has now to make 9(y 2 + 1)/16 a square, to do this he assumes that 9y 2 + 9 = (3y − 4)2 , hence y = 7/24. Therefore the squares required are 9/16 and 49/576. It will be recollected that Diophantus had only one symbol for an unknown quantity; and in this example he begins by calling the unknowns x2 and 1, but as soon as he has found x he then replaces the 1 by the symbol for the unknown quantity, and finds it in its turn. (iv) To find a [rational ] right-angled triangle such that the line bisecting an acute angle is rational [book vi, prob. 18]. A

B

D

C

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His solution is as follows. Let ABC be the triangle of which C is the right-angle. Let the bisector AD = 5x, and let DC = 3x, hence AC = 4x. Next let BC be a multiple of 3, say 3, ∴ BD = 3−3x, hence AB = 4 − 4x (by Euc. vi, 3). Hence (4 − 4x)2 = 32 + (4x)2 (Euc. i, 47), ∴ x = 7/32. Multiplying by 32 we get for the sides of the triangle 28, 96, and 100; and for the bisector 35. (v) A man buys x measures of wine, some at 8 drachmae a measure, the rest at 5. He pays for them a square number of drachmae, such that, if 60 be added to it, the resulting number is x2 . Find the number he bought at each price [book v, prob. 33]. The price paid was x2 − 60, hence 8x > x2 − 60 and 5x < x2 − 60. From this it follows that x must be greater than 11 and less than 12. Again x2 − 60 is to be a square; suppose it is equal to (x − m)2 then x = (m2 + 60)/2m, we have therefore m2 + 60 < 12; 2m ∴ 19 < m < 21.

11

and < to represent greater than and less than. When he denoted the unknown quantity by a he represented a2 by aa, a3 by aaa, and so on. This is a distinct improvement on Vieta’s notation. The same symbolism was used by Wallis as late as 1685, but concurrently with the modern index notation which was introduced by Descartes. I need not allude to the other investigations of Harriot, as they are comparatively of small importance; extracts from some of them were published by S. P. Rigaud in 1833. Oughtred. Among those who contributed to the general adoption in England of these various improvements and additions to algorism and algebra was William Oughtred,1 who was born at Eton on March 5, 1574, and died at his vicarage of Albury in Surrey on June 30, 1660: it is sometimes said that the cause of his death was the excitement and delight which he experienced “at hearing the House of Commons [or Convention] had voted the King’s return”; a recent critic adds that it should be remembered “by way of excuse that he [Oughtred] was then eighty-six years old,” but perhaps the story is sufficiently discredited by the date of his death. Oughtred was educated at Eton and King’s College, Cambridge, of the latter of which colleges he was a fellow and for some time mathematical lecturer. His Clavis Mathematicae published in 1631 is a good systematic text-book on arithmetic, and it contains practically all that was then known on the subject. In this work he introduced the symbol × for multiplication. He also introduced the symbol : : in proportion: previously to his time a proportion such as a : b = c : d was usually written 1

See William Oughtred, by F. Cajori, Chicago, 1916. A complete edition of Oughtred’s works was published at Oxford in 1677.

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as a − b − c − d; he denoted it by a . b : : c . d. Wallis says that some found fault with the book on account of the style, but that they only displayed their own incompetence, for Oughtred’s “words be always full but not redundant.” Pell makes a somewhat similar remark. Oughtred also wrote a treatise on trigonometry published in 1657, in which abbreviations for sine, cosine, &c., were employed. This was really an important advance, but the works of Girard and Oughtred, in which they were used, were neglected and soon forgotten, and it was not until Euler reintroduced contractions for the trigonometrical functions that they were generally adopted. In this work the colon (i.e. the symbol :) was used to denote a ratio. We may say roughly that henceforth elementary arithmetic, algebra, and trigonometry were treated in a manner which is not substantially different from that now in use; and that the subsequent improvements introduced were additions to the subjects as then known, and not a rearrangement of them on new foundations. The origin of the more common symbols in algebra. It may be convenient if I collect here in one place the scattered remarks I have made on the introduction of the various symbols for the more common operations in algebra.1 The later Greeks, the Hindoos, and Jordanus indicated addition by mere juxtaposition. It will be observed that this is still the custom in arithmetic, where, for instance, 2 12 stands for 2 + 21 . The Italian algebraists, when they gave up expressing every operation in words at full length and introduced syncopated algebra, usually denoted plus by its initial letter P or p, a line being sometimes drawn through the letter to show that it was a contraction, or a symbol of operation, and not a quantity. The practice, however, was not uniform; Pacioli, for example, sometimes denoted plus by p¯, and sometimes by e, and Tartaglia commonly denoted it by φ. The German and English algebraists, on the other hand, introduced the sign + almost as soon as they used algorism, but they spoke of it as signum additorum and employed it only to indicate excess; they also used it with a special meaning in solutions by the method of false assumption. Widman used it as an abbreviation 1

See also two articles by C. Henry in the June and July numbers of the Revue Arch´eologique, 1879, vol. xxxvii, pp. 324–333, vol. xxxviii, pp. 1–10.

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for excess in 1489: by 1630 it was part of the recognised notation of algebra, and was used as a symbol of operation. Subtraction was indicated by Diophantus by an inverted and truncated ψ. The Hindoos denoted it by a dot. The Italian algebraists when they introduced syncopated algebra generally denoted minus by M or m, a line being sometimes drawn through the letter; but the practice was not uniform—Pacioli, for example, denoting it sometimes by m, ¯ and sometimes by de for demptus. The German and English algebraists introduced the present symbol which they described as signum subtractorum. It is most likely that the vertical bar in the symbol for plus was superimposed on the symbol for minus to distinguish the two. It may be noticed that Pacioli and Tartaglia found the sign − already used to denote a division, a ratio, or a proportion indifferently. The present sign for minus was in general use by about the year 1630, and was then employed as a symbol of operation. Vieta, Schooten, and others among their contemporaries employed the sign = written between two quantities to denote the difference between them; thus a = b means with them what we denote by a ∼ b. On the other hand, Barrow wrote −−: for the same purpose. I am not aware when or by whom the current symbol ∼ was first used with this signification. Oughtred in 1631 used the sign × to indicate multiplication; Harriot in 1631 denoted the operation by a dot; Descartes in 1637 indicated it by juxtaposition. I am not aware of any symbols for it which were in previous use. Leibnitz in 1686 employed the sign _ to denote multiplication. Division was ordinarily denoted by the Arab way of writing the quantities in the form of a fraction by means of a line drawn between a them in any of the forms a − b, a/b, or . Oughtred in 1631 employed b a dot to denote either division or a ratio. Leibnitz in 1686 employed the sign ^ to denote division. The colon (or symbol :), used to denote a ratio, occurs on the last two pages of Oughtred’s Canones Sinuum, published in 1657. I believe that the current symbol for division ÷ is only a combination of the − and the symbol : for a ratio; it was used by Johann Heinrich Rahn at Z¨ urich in 1659, and by John Pell in London in 1668. The symbol ÷÷ was used by Barrow and other writers of his time to indicate continued proportion. The current symbol for equality was introduced by Record in 1557; Xylander in 1575 denoted it by two parallel vertical lines; but in general

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8

8

till the year 1600 the word was written at length; and from then until the time of Newton, say about 1680, it was more frequently represented by or by than by any other symbol. Either of these latter signs was used as a contraction for the first two letters of the word aequalis. The symbol : : to denote proportion, or the equality of two ratios, was introduced by Oughtred in 1631, and was brought into common use by Wallis in 1686. There is no object in having a symbol to indicate the equality of two ratios which is different from that used to indicate the equality of other things, and it is better to replace it by the sign =. The sign > for is greater than and the sign < for is less than were introduced by Harriot in 1631, but Oughtred simultaneously invented and for the same purpose; and these latter were the symbols frequently used till the beginning of the eighteenth century, ex. gr. by Barrow. The symbols =| for is not equal to, > | for is not greater than, and

Title: A Short Account of the History of Mathematics Author: W. W. Rouse Ball Release Date: May 28, 2010 [EBook #31246] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICS ***

A SHORT ACCOUNT OF THE

HISTORY OF MATHEMATICS

BY

W. W. ROUSE BALL FELLOW OF TRINITY COLLEGE, CAMBRIDGE

DOVER PUBLICATIONS, INC. NEW YORK

This new Dover edition, first published in 1960, is an unabridged and unaltered republication of the author’s last revision—the fourth edition which appeared in 1908.

International Standard Book Number: 0-486-20630-0 Library of Congress Catalog Card Number: 60-3187

Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N. Y. 10014

Produced by Greg Lindahl, Viv, Juliet Sutherland, Nigel Blower and the Online Distributed Proofreading Team at http://www.pgdp.net

Transcriber’s Notes A small number of minor typographical errors and inconsistencies have been corrected. References to figures such as “on the next page” have been replaced with text such as “below” which is more suited to an eBook. Such changes are documented in the LATEX source: %[**TN: text of note]

PREFACE. The subject-matter of this book is a historical summary of the development of mathematics, illustrated by the lives and discoveries of those to whom the progress of the science is mainly due. It may serve as an introduction to more elaborate works on the subject, but primarily it is intended to give a short and popular account of those leading facts in the history of mathematics which many who are unwilling, or have not the time, to study it systematically may yet desire to know. The first edition was substantially a transcript of some lectures which I delivered in the year 1888 with the object of giving a sketch of the history, previous to the nineteenth century, that should be intelligible to any one acquainted with the elements of mathematics. In the second edition, issued in 1893, I rearranged parts of it, and introduced a good deal of additional matter. The scheme of arrangement will be gathered from the table of contents at the end of this preface. Shortly it is as follows. The first chapter contains a brief statement of what is known concerning the mathematics of the Egyptians and Phoenicians; this is introductory to the history of mathematics under Greek influence. The subsequent history is divided into three periods: first, that under Greek influence, chapters ii to vii; second, that of the middle ages and renaissance, chapters viii to xiii; and lastly that of modern times, chapters xiv to xix. In discussing the mathematics of these periods I have confined myself to giving the leading events in the history, and frequently have passed in silence over men or works whose influence was comparatively unimportant. Doubtless an exaggerated view of the discoveries of those mathematicians who are mentioned may be caused by the non-allusion to minor writers who preceded and prepared the way for them, but in all historical sketches this is to some extent inevitable, and I have done my best to guard against it by interpolating remarks on the progress

PREFACE

v

of the science at different times. Perhaps also I should here state that generally I have not referred to the results obtained by practical astronomers and physicists unless there was some mathematical interest in them. In quoting results I have commonly made use of modern notation; the reader must therefore recollect that, while the matter is the same as that of any writer to whom allusion is made, his proof is sometimes translated into a more convenient and familiar language. The greater part of my account is a compilation from existing histories or memoirs, as indeed must be necessarily the case where the works discussed are so numerous and cover so much ground. When authorities disagree I have generally stated only that view which seems to me to be the most probable; but if the question be one of importance, I believe that I have always indicated that there is a difference of opinion about it. I think that it is undesirable to overload a popular account with a mass of detailed references or the authority for every particular fact mentioned. For the history previous to 1758, I need only refer, once for all, to the closely printed pages of M. Cantor’s monumental Vorlesungen u ¨ber die Geschichte der Mathematik (hereafter alluded to as Cantor), which may be regarded as the standard treatise on the subject, but usually I have given references to the other leading authorities on which I have relied or with which I am acquainted. My account for the period subsequent to 1758 is generally based on the memoirs or monographs referred to in the footnotes, but the main facts to 1799 have been also enumerated in a supplementary volume issued by Prof. Cantor last year. I hope that my footnotes will supply the means of studying in detail the history of mathematics at any specified period should the reader desire to do so. My thanks are due to various friends and correspondents who have called my attention to points in the previous editions. I shall be grateful for notices of additions or corrections which may occur to any of my readers.

W. W. ROUSE BALL. TRINITY COLLEGE, CAMBRIDGE.

NOTE. The fourth edition was stereotyped in 1908, but no material changes have been made since the issue of the second edition in 1893, other duties having, for a few years, rendered it impossible for me to find time for any extensive revision. Such revision and incorporation of recent researches on the subject have now to be postponed till the cost of printing has fallen, though advantage has been taken of reprints to make trivial corrections and additions.

W. W. R. B. TRINITY COLLEGE, CAMBRIDGE.

vi

vii

TABLE OF CONTENTS. Preface . . . Table of Contents

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

page iv vii

Chapter I. Egyptian and Phoenician Mathematics. The history of mathematics begins with that of the Ionian Greeks Greek indebtedness to Egyptians and Phoenicians . . . Knowledge of the science of numbers possessed by the Phoenicians Knowledge of the science of numbers possessed by the Egyptians Knowledge of the science of geometry possessed by the Egyptians Note on ignorance of mathematics shewn by the Chinese . .

. . . . . .

. . . . . .

1 1 2 2 4 7

First Period. Mathematics under Greek Influence. This period begins with the teaching of Thales, circ. 600 b.c., and ends with the capture of Alexandria by the Mohammedans in or about 641 a.d. The characteristic feature of this period is the development of geometry.

Chapter II. The Ionian and Pythagorean Schools. Circ. 600 b.c.–400 b.c. Authorities . . . . . . . . The Ionian School . . . . . . . Thales, 640–550 b.c. . . . . . . His geometrical discoveries . . . . His astronomical teaching . . . . . Anaximander. Anaximenes. Mamercus. Mandryatus The Pythagorean School . . . . . . Pythagoras, 569–500 b.c. . . . . . The Pythagorean teaching . . . . . The Pythagorean geometry . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

10 11 11 11 13 14 15 15 15 17

viii

TABLE OF CONTENTS The Pythagorean theory of numbers . . . . Epicharmus. Hippasus. Philolaus. Archippus. Lysis . . Archytas, circ. 400 b.c. . . . . . . . His solution of the duplication of a cube . . . Theodorus. Timaeus. Bryso . . . . . . Other Greek Mathematical Schools in the Fifth Century b.c. Oenopides of Chios . . . . . . . . Zeno of Elea. Democritus of Abdera . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

19 22 22 23 24 24 24 25

Chapter III. The Schools of Athens and Cyzicus. Circ. 420–300 b.c. Authorities . . . . . . . . . Mathematical teachers at Athens prior to 420 b.c. . . Anaxagoras. The Sophists. Hippias (The quadratrix). Antipho . . . . . . . . . Three problems in which these schools were specially interested Hippocrates of Chios, circ. 420 b.c. . . . . Letters used to describe geometrical diagrams . . Introduction in geometry of the method of reduction The quadrature of certain lunes . . . . . The problem of the duplication of the cube . . Plato, 429–348 b.c. . . . . . . . . Introduction in geometry of the method of analysis . Theorem on the duplication of the cube . . . Eudoxus, 408–355 b.c. . . . . . . . Theorems on the golden section . . . . . Introduction of the method of exhaustions . . Pupils of Plato and Eudoxus . . . . . . Menaechmus, circ. 340 b.c. . . . . . . Discussion of the conic sections . . . . . His two solutions of the duplication of the cube . Aristaeus. Theaetetus . . . . . . . Aristotle, 384–322 b.c. . . . . . . . Questions on mechanics. Letters used to indicate magnitudes

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27 27 27 29 30 31 31 32 32 34 34 35 36 36 36 37 38 38 38 38 39 39 40

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41 41 43 43 44 44 47

Chapter IV. The First Alexandrian School. Circ. 300–30 b.c. Authorities . . . . . . . . . Foundation of Alexandria . . . . . . . The Third Century before Christ . . . . . Euclid, circ. 330–275 b.c. . . . . . . Euclid’s Elements . . . . . . . The Elements as a text-book of geometry . . . The Elements as a text-book of the theory of numbers

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TABLE OF CONTENTS Euclid’s other works . . . . . . . Aristarchus, circ. 310–250 b.c. . . . . . . Method of determining the distance of the sun . . Conon. Dositheus. Zeuxippus. Nicoteles . . . . Archimedes, 287–212 b.c. . . . . . . His works on plane geometry . . . . . His works on geometry of three dimensions . . His two papers on arithmetic, and the “cattle problem” His works on the statics of solids and fluids . . His astronomy . . . . . . . . The principles of geometry assumed by Archimedes . Apollonius, circ. 260–200 b.c. . . . . . His conic sections . . . . . . . His other works . . . . . . . . His solution of the duplication of the cube . . Contrast between his geometry and that of Archimedes Eratosthenes, 275–194 b.c. . . . . . . The Sieve of Eratosthenes . . . . . . The Second Century before Christ . . . . . Hypsicles (Euclid, book xiv). Nicomedes. Diocles . . Perseus. Zenodorus . . . . . . . . Hipparchus, circ. 130 b.c. . . . . . . Foundation of scientific astronomy . . . . Foundation of trigonometry . . . . . Hero of Alexandria, circ. 125 b.c. . . . . . Foundation of scientific engineering and of land-surveying Area of a triangle determined in terms of its sides . Features of Hero’s works . . . . . . The First Century before Christ . . . . . Theodosius . . . . . . . . . Dionysodorus . . . . . . . . . End of the First Alexandrian School . . . . . Egypt constituted a Roman province . . . .

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49 51 51 52 53 55 58 59 60 63 63 63 64 66 67 68 69 69 70 70 71 71 72 73 73 73 74 75 76 76 76 76 76

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78 79 79 79 79 80 80 80 80 80

Chapter V. The Second Alexandrian School. 30 b.c.–641 a.d. Authorities . . . . . . . . . The First Century after Christ . . . . . . Serenus. Menelaus . . . . . . . . Nicomachus . . . . . . . . . Introduction of the arithmetic current in medieval Europe The Second Century after Christ . . . . . Theon of Smyrna. Thymaridas . . . . . . Ptolemy, died in 168 . . . . . . . The Almagest . . . . . . . . Ptolemy’s astronomy . . . . . . .

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TABLE OF CONTENTS Ptolemy’s geometry . . . . . . . The Third Century after Christ . . . . . Pappus, circ. 280 . . . . . . . . The Συναγωγή, a synopsis of Greek mathematics . The Fourth Century after Christ . . . . . Metrodorus. Elementary problems in arithmetic and algebra Three stages in the development of algebra . . . Diophantus, circ. 320 (?) . . . . . . Introduction of syncopated algebra in his Arithmetic The notation, methods, and subject-matter of the work His Porisms . . . . . . . . Subsequent neglect of his discoveries . . . . Iamblichus . . . . . . . . . Theon of Alexandria. Hypatia . . . . . . Hostility of the Eastern Church to Greek science . . The Athenian School (in the Fifth Century) . . . Proclus, 412–485. Damascius. Eutocius . . . . Roman Mathematics . . . . . . . . Nature and extent of the mathematics read at Rome . Contrast between the conditions at Rome and at Alexandria End of the Second Alexandrian School . . . . The capture of Alexandria, and end of the Alexandrian Schools

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82 83 83 83 85 85 86 86 87 87 91 92 92 92 93 93 93 94 94 95 96 96

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97 97 99 100

Chapter VI. The Byzantine School. 641–1453. Preservation of works of the great Greek Mathematicians . Hero of Constantinople. Psellus. Planudes. Barlaam. Argyrus . Nicholas Rhabdas, Pachymeres. Moschopulus (Magic Squares) Capture of Constantinople, and dispersal of Greek Mathematicians

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Chapter VII. Systems of Numeration and Primitive Arithmetic. Authorities . . . . . . . . . . Methods of counting and indicating numbers among primitive races Use of the abacus or swan-pan for practical calculation . . Methods of representing numbers in writing . . . . The Roman and Attic symbols for numbers . . . . The Alexandrian (or later Greek) symbols for numbers . . Greek arithmetic . . . . . . . . . Adoption of the Arabic system of notation among civilized races

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101 101 103 105 105 106 106 107

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Second Period. Mathematics of the Middle Ages and of the Renaissance. This period begins about the sixth century, and may be said to end with the invention of analytical geometry and of the infinitesimal calculus. The characteristic feature of this period is the creation or development of modern arithmetic, algebra, and trigonometry.

Chapter VIII. The Rise Of Learning In Western Europe. Circ. 600–1200. Authorities . . . . . . . . Education in the Sixth, Seventh, and Eighth Centuries The Monastic Schools . . . . . . Boethius, circ. 475–526 . . . . . . Medieval text-books in geometry and arithmetic Cassiodorus, 490–566. Isidorus of Seville, 570–636 . The Cathedral and Conventual Schools . . . The Schools of Charles the Great . . . . Alcuin, 735–804 . . . . . . . Education in the Ninth and Tenth Centuries . . Gerbert (Sylvester II.), died in 1003 . . . . Bernelinus . . . . . . . . The Early Medieval Universities . . . . Rise during the twelfth century of the earliest universities Development of the medieval universities . . . Outline of the course of studies in a medieval university

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109 109 109 110 110 111 111 111 111 113 113 115 115 115 116 117

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120 120 121 121 122 122 123 123 125 125 127 129 129 130 130

Chapter IX. The Mathematics Of The Arabs. Authorities . . . . . . . . Extent of Mathematics obtained from Greek Sources . The College of Scribes . . . . . . Extent of Mathematics obtained from the (Aryan) Hindoos Arya-Bhata, circ. 530 . . . . . . His algebra and trigonometry (in his Aryabhathiya) Brahmagupta, circ. 640 . . . . . . His algebra and geometry (in his Siddhanta) . Bhaskara, circ. 1140 . . . . . . The Lilavati or arithmetic; decimal numeration used The Bija Ganita or algebra . . . . Development of Mathematics in Arabia . . . ¯ rizm¯i, circ. 830 . Alkarismi or Al-Khwa . . His Al-gebr we’ l mukabala . . . . His solution of a quadratic equation . . .

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TABLE OF CONTENTS Introduction of Arabic or Indian system of numeration Tabit ibn Korra, 836–901; solution of a cubic equation Alkayami. Alkarki. Development of algebra . . . Albategni. Albuzjani. Development of trigonometry . . Alhazen. Abd-al-gehl. Development of geometry . . Characteristics of the Arabian School . . . .

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131 132 132 133 134 134

Chapter X. Introduction of Arabian Works into Europe. Circ. 1150–1450. The Eleventh Century . . . . . . . Moorish Teachers. Geber ibn Aphla. Arzachel . . . The Twelfth Century . . . . . . . Adelhard of Bath . . . . . . . . Ben-Ezra. Gerard. John Hispalensis . . . . . The Thirteenth Century . . . . . . . Leonardo of Pisa, circ. 1175–1230 . . . . The Liber Abaci, 1202 . . . . . . The introduction of the Arabic numerals into commerce The introduction of the Arabic numerals into science The mathematical tournament . . . . . Frederick II., 1194–1250 . . . . . . . Jordanus, circ. 1220 . . . . . . . His De Numeris Datis; syncopated algebra . . Holywood . . . . . . . . . . Roger Bacon, 1214–1294 . . . . . . Campanus . . . . . . . . . The Fourteenth Century . . . . . . . Bradwardine . . . . . . . . . Oresmus . . . . . . . . . . The reform of the university curriculum . . . . The Fifteenth Century . . . . . . . Beldomandi . . . . . . . . .

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136 136 137 137 137 138 138 138 139 139 140 141 141 142 144 144 147 147 147 147 148 149 149

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151 151 151 152 154 156 156 162

Chapter XI. The Development Of Arithmetic. Circ. 1300–1637. Authorities . . . . . . . . . The Boethian arithmetic . . . . . . . Algorism or modern arithmetic . . . . . . The Arabic (or Indian) symbols: history of . . . Introduction into Europe by science, commerce, and calendars Improvements introduced in algoristic arithmetic . . (i) Simplification of the fundamental processes . . (ii) Introduction of signs for addition and subtraction

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TABLE OF CONTENTS (iii) Invention of logarithms, 1614 (iv) Use of decimals, 1619 . .

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162 163

Chapter XII. The Mathematics of the Renaissance. Circ. 1450–1637. Authorities . . . . . . . . . Effect of invention of printing. The renaissance . . Development of Syncopated Algebra and Trigonometry . Regiomontanus, 1436–1476 . . . . . . His De Triangulis (printed in 1496) . . . . Purbach, 1423–1461. Cusa, 1401–1464. Chuquet, circ. 1484 Introduction and origin of symbols + and − . . . Pacioli or Lucas di Burgo, circ. 1500 . . . . His arithmetic and geometry, 1494 . . . . Leonardo da Vinci, 1452–1519 . . . . . . D¨ urer, 1471–1528. Copernicus, 1473–1543 . . . Record, 1510–1558; introduction of symbol for equality . Rudolff, circ. 1525. Riese, 1489–1559 . . . . Stifel, 1486–1567 . . . . . . . . His Arithmetica Integra, 1544 . . . . . Tartaglia, 1500–1557 . . . . . . . His solution of a cubic equation, 1535 . . . His arithmetic, 1556–1560 . . . . . . Cardan, 1501–1576 . . . . . . . . His Ars Magna, 1545; the third work printed on algebra. His solution of a cubic equation . . . . . Ferrari, 1522–1565; solution of a biquadratic equation . Rheticus, 1514–1576. Maurolycus. Borrel. Xylander . Commandino. Peletier. Romanus. Pitiscus. Ramus. 1515–1572 Bombelli, circ. 1570 . . . . . . . . Development of Symbolic Algebra . . . . . Vieta, 1540–1603 . . . . . . . . The In Artem; introduction of symbolic algebra, 1591 Vieta’s other works . . . . . . . Girard, 1595–1632; development of trigonometry and algebra Napier, 1550–1617; introduction of logarithms, 1614 . Briggs, 1561–1631; calculations of tables of logarithms . Harriot, 1560–1621; development of analysis in algebra . Oughtred, 1574–1660 . . . . . . . The Origin of the more Common Symbols in Algebra .

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165 165 166 166 167 170 171 173 173 176 176 177 178 178 179 180 181 182 183 184 186 186 187 187 188 189 189 191 192 194 195 196 196 197 198

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Chapter XIII. The Close of the Renaissance. Circ. 1586–1637. Authorities . . . . . . . . . Development of Mechanics and Experimental Methods . Stevinus, 1548–1620 . . . . . . . Commencement of the modern treatment of statics, 1586 Galileo, 1564–1642 . . . . . . . . Commencement of the science of dynamics . . Galileo’s astronomy . . . . . . . Francis Bacon, 1561–1626. Guldinus, 1577–1643 . . Wright, 1560–1615; construction of maps . . . . Snell, 1591–1626 . . . . . . . . Revival of Interest in Pure Geometry . . . . Kepler, 1571–1630 . . . . . . . . His Paralipomena, 1604; principle of continuity . . His Stereometria, 1615; use of infinitesimals . . Kepler’s laws of planetary motion, 1609 and 1619 . Desargues, 1593–1662 . . . . . . . His Brouillon project; use of projective geometry . Mathematical Knowledge at the Close of the Renaissance .

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202 202 202 203 205 205 206 208 209 210 210 210 211 212 212 213 213 214

Third period. Modern Mathematics. This period begins with the invention of analytical geometry and the infinitesimal calculus. The mathematics is far more complex than that produced in either of the preceding periods: but it may be generally described as characterized by the development of analysis, and its application to the phenomena of nature.

Chapter XIV. The History of Modern Mathematics. Treatment of the subject . . . . . . . . Invention of analytical geometry and the method of indivisibles Invention of the calculus . . . . . . . . Development of mechanics . . . . . . . Application of mathematics to physics . . . . . Recent development of pure mathematics . . . . .

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217 218 218 219 219 220

Chapter XV. History of Mathematics from Descartes to Huygens. Circ. 1635–1675. Authorities . . . Descartes, 1596–1650 . His views on philosophy

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221 221 224

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TABLE OF CONTENTS His invention of analytical geometry, 1637 . . . . . His algebra, optics, and theory of vortices . . . . . Cavalieri, 1598–1647 . . . . . . . . . The method of indivisibles . . . . . . . . Pascal, 1623–1662 . . . . . . . . . . His geometrical conics . . . . . . . . The arithmetical triangle . . . . . . . . Foundation of the theory of probabilities, 1654 . . . . His discussion of the cycloid . . . . . . . Wallis, 1616–1703 . . . . . . . . . . The Arithmetica Infinitorum, 1656 . . . . . . Law of indices in algebra . . . . . . . . Use of series in quadratures . . . . . . . Earliest rectification of curves, 1657 . . . . . . Wallis’s algebra . . . . . . . . . . Fermat, 1601–1665 . . . . . . . . . . His investigations on the theory of numbers . . . . His use in geometry of analysis and of infinitesimals . . . Foundation of the theory of probabilities, 1654 . . . . Huygens, 1629–1695 . . . . . . . . . The Horologium Oscillatorium, 1673 . . . . . . The undulatory theory of light . . . . . . . Other Mathematicians of this Time . . . . . . . Bachet . . . . . . . . . . . . Mersenne; theorem on primes and perfect numbers . . . . Roberval. Van Schooten. Saint-Vincent . . . . . . Torricelli. Hudde. Fr´enicle . . . . . . . . De Laloub`ere. Mercator. Barrow; the differential triangle . . Brouncker; continued fractions . . . . . . . . James Gregory; distinction between convergent and divergent series Sir Christopher Wren . . . . . . . . . Hooke. Collins . . . . . . . . . . . Pell. Sluze. Viviani . . . . . . . . . . Tschirnhausen. De la Hire. Roemer. Rolle . . . . .

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224 227 229 230 232 234 234 235 236 237 238 238 239 240 241 241 242 246 247 248 249 250 251 252 252 253 254 254 257 258 259 259 260 261

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263 263 264 265 266 267 267 268 269 272

Chapter XVI. The Life and Works of Newton. Authorities . . . . . . Newton’s school and undergraduate life . Investigations in 1665–1666 on fluxions, optics, His views on gravitation, 1666 . . Researches in 1667–1669 . . . . Elected Lucasian professor, 1669 . . Optical lectures and discoveries, 1669–1671 Emission theory of light, 1675 . . . The Leibnitz Letters, 1676 . . . Discoveries on gravitation, 1679 . .

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TABLE OF CONTENTS Discoveries and lectures on algebra, 1673–1683 . . Discoveries and lectures on gravitation, 1684 . . The Principia, 1685–1686 . . . . . . The subject-matter of the Principia . . . Publication of the Principia . . . . Investigations and work from 1686 to 1696 . . Appointment at the Mint, and removal to London, 1696 Publication of the Optics, 1704 . . . . . Appendix on classification of cubic curves . . Appendix on quadrature by means of infinite series Appendix on method of fluxions . . . The invention of fluxions and the infinitesimal calculus Newton’s death, 1727 . . . . . . List of his works . . . . . . . Newton’s character . . . . . . . Newton’s discoveries . . . . . . .

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272 274 275 276 278 278 279 279 279 281 282 286 286 286 287 289

Chapter XVII. Leibnitz and the Mathematicians of the First Half of the Eighteenth Century. Authorities . . . . . . . . . . Leibnitz and the Bernoullis . . . . . . . Leibnitz, 1646–1716 . . . . . . . . His system of philosophy, and services to literature . . The controversy as to the origin of the calculus . . . His memoirs on the infinitesimal calculus . . . . His papers on various mechanical problems . . . Characteristics of his work . . . . . . . James Bernoulli, 1654–1705 . . . . . . . John Bernoulli, 1667–1748 . . . . . . . The younger Bernouillis . . . . . . . . Development of Analysis on the Continent . . . . L’Hospital, 1661–1704 . . . . . . . . Varignon, 1654–1722. De Montmort. Nicole . . . . Parent. Saurin. De Gua. Cramer, 1704–1752 . . . . Riccati, 1676–1754. Fagnano, 1682–1766 . . . . . Clairaut, 1713–1765 . . . . . . . . D’Alembert, 1717–1783 . . . . . . . . Solution of a partial differential equation of the second order Daniel Bernoulli, 1700–1782 . . . . . . . English Mathematicians of the Eighteenth Century . . . David Gregory, 1661–1708. Halley, 1656–1742 . . . . Ditton, 1675–1715 . . . . . . . . . Brook Taylor, 1685–1731 . . . . . . . Taylor’s theorem . . . . . . . . Taylor’s physical researches . . . . . . Cotes, 1682–1716 . . . . . . . . .

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291 291 291 293 293 298 299 301 301 302 303 304 304 305 305 306 307 308 309 311 312 312 313 313 314 314 315

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TABLE OF CONTENTS Demoivre, 1667–1754; development of trigonometry Maclaurin, 1698–1746 . . . . . His geometrical discoveries . . . The Treatise of Fluxions . . . . His propositions on attractions . . . Stewart, 1717–1785. Thomas Simpson, 1710–1761

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315 316 317 318 318 319

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322 323 323 324 326 326 326 327 329 330 330 331 334 337 338 338 339 339 340 340 341 341 342 343 344 344 345 346 346 347 348 348 349 349 350 350 351

Chapter XVIII. Lagrange, Laplace, and their Contemporaries. Circ. 1740–1830. Characteristics of the mathematics of the period . . . Development of Analysis and Mechanics . . . . . Euler, 1707–1783 . . . . . . . . . The Introductio in Analysin Infinitorum, 1748 . . . The Institutiones Calculi Differentialis, 1755 . . . The Institutiones Calculi Integralis, 1768–1770 . . . The Anleitung zur Algebra, 1770 . . . . . Euler’s works on mechanics and astronomy . . . Lambert, 1728–1777 . . . . . . . . . B´ezout, 1730–1783. Trembley, 1749–1811. Arbogast, 1759–1803 Lagrange, 1736–1813 . . . . . . . . Memoirs on various subjects . . . . . . The M´ecanique analytique, 1788 . . . . . The Th´eorie and Calcul des fonctions, 1797, 1804 . . The R´esolution des ´equations num´eriques, 1798. . . Characteristics of Lagrange’s work . . . . . Laplace, 1749–1827 . . . . . . . . Memoirs on astronomy and attractions, 1773–1784 . . Use of spherical harmonics and the potential . . . Memoirs on problems in astronomy, 1784–1786 . . . The M´ecanique c´eleste and Exposition du syst`eme du monde The Nebular Hypothesis . . . . . . . The Meteoric Hypothesis . . . . . . . The Th´eorie analytique des probabilit´es, 1812 . . . The Method of Least Squares . . . . . . Other researches in pure mathematics and in physics . Characteristics of Laplace’s work . . . . . Character of Laplace . . . . . . . . Legendre, 1752–1833 . . . . . . . . His memoirs on attractions . . . . . . The Th´eorie des nombres, 1798 . . . . . . Law of quadratic reciprocity . . . . . . The Calcul int´egral and the Fonctions elliptiques . . Pfaff, 1765–1825 . . . . . . . . . Creation of Modern Geometry . . . . . . . Monge, 1746–1818 . . . . . . . . . Lazare Carnot, 1753–1823. Poncelet, 1788–1867 . . .

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TABLE OF CONTENTS Development of Mathematical Physics . . . Cavendish, 1731–1810 . . . . . . Rumford, 1753–1815. Young, 1773–1829 . . . Dalton, 1766–1844 . . . . . . . Fourier, 1768–1830 . . . . . . . Sadi Carnot; foundation of thermodynamics . . Poisson, 1781–1840 . . . . . . . Amp`ere, 1775–1836. Fresnel, 1788–1827. Biot, 1774–1862 Arago, 1786–1853 . . . . . . . Introduction of Analysis into England . . . Ivory, 1765–1842 . . . . . . . The Cambridge Analytical School . . . . Woodhouse, 1773–1827 . . . . . . Peacock, 1791–1858. Babbage, 1792–1871. John Herschel,

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353 353 353 354 355 356 356 358 359 360 360 361 361 362

Chapter XIX. Mathematics of the Nineteenth Century. Creation of new branches of mathematics . . . . . Difficulty in discussing the mathematics of this century . . Account of contemporary work not intended to be exhaustive . Authorities . . . . . . . . . . Gauss, 1777–1855 . . . . . . . . . Investigations in astronomy . . . . . . Investigations in electricity . . . . . . The Disquisitiones Arithmeticae, 1801 . . . . His other discoveries . . . . . . . . Comparison of Lagrange, Laplace, and Gauss . . . Dirichlet, 1805–1859 . . . . . . . . . Development of the Theory of Numbers . . . . . Eisenstein, 1823–1852 . . . . . . . . Henry Smith, 1826–1883 . . . . . . . . Kummer, 1810–1893 . . . . . . . . . Notes on other writers on the Theory of Numbers . . . Development of the Theory of Functions of Multiple Periodicity Abel, 1802–1829. Abel’s Theorem . . . . . . Jacobi, 1804–1851 . . . . . . . . . Riemann, 1826–1866 . . . . . . . . Notes on other writers on Elliptic and Abelian Functions . . Weierstrass, 1815–1897 . . . . . . . Notes on recent writers on Elliptic and Abelian Functions . The Theory of Functions . . . . . . . . Development of Higher Algebra . . . . . . Cauchy, 1789–1857 . . . . . . . . . Argand, 1768–1822; geometrical interpretation of complex numbers Sir William Hamilton, 1805–1865; introduction of quaternions Grassmann, 1809–1877; his non-commutative algebra, 1844 . Boole, 1815–1864. De Morgan, 1806–1871 . . . .

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365 365 365 366 367 368 369 371 372 373 373 374 374 374 377 377 378 379 380 381 382 382 383 384 385 385 387 387 389 389

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TABLE OF CONTENTS Galois, 1811–1832; theory of discontinuous substitution groups Cayley, 1821–1895 . . . . . . . . . Sylvester, 1814–1897 . . . . . . . . Lie, 1842–1889; theory of continuous substitution groups . . Hermite, 1822–1901 . . . . . . . . Notes on other writers on Higher Algebra . . . . . Development of Analytical Geometry . . . . . Notes on some recent writers on Analytical Geometry . . Line Geometry . . . . . . . . . . Analysis. Names of some recent writers on Analysis . . . Development of Synthetic Geometry . . . . . . Steiner, 1796–1863 . . . . . . . . . Von Staudt, 1798–1867 . . . . . . . . Other writers on modern Synthetic Geometry . . . . Development of Non-Euclidean Geometry . . . . . Euclid’s Postulate on Parallel Lines . . . . . Hyperbolic Geometry. Elliptic Geometry . . . . Congruent Figures . . . . . . . . Foundations of Mathematics. Assumptions made in the subject Kinematics . . . . . . . . . . Development of the Theory of Mechanics, treated Graphically . Development of Theoretical Mechanics, treated Analytically . Notes on recent writers on Mechanics . . . . . Development of Theoretical Astronomy . . . . . Bessel, 1784–1846 . . . . . . . . . Leverrier, 1811–1877. Adams, 1819–1892 . . . . . Notes on other writers on Theoretical Astronomy . . . Recent Developments . . . . . . . . Development of Mathematical Physics . . . . .

Index

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390 390 391 392 392 393 395 395 396 396 397 397 398 398 398 399 399 401 402 402 402 403 405 405 405 406 407 408 409

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410

1

CHAPTER I. egyptian and phoenician mathematics. The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks. The subsequent history may be divided into three periods, the distinctions between which are tolerably well marked. The first period is that of the history of mathematics under Greek influence, this is discussed in chapters ii to vii; the second is that of the mathematics of the middle ages and the renaissance, this is discussed in chapters viii to xiii; the third is that of modern mathematics, and this is discussed in chapters xiv to xix. Although the history of mathematics commences with that of the Ionian schools, there is no doubt that those Greeks who first paid attention to the subject were largely indebted to the previous investigations of the Egyptians and Phoenicians. Our knowledge of the mathematical attainments of those races is imperfect and partly conjectural, but, such as it is, it is here briefly summarised. The definite history begins with the next chapter. On the subject of prehistoric mathematics, we may observe in the first place that, though all early races which have left records behind them knew something of numeration and mechanics, and though the majority were also acquainted with the elements of land-surveying, yet the rules which they possessed were in general founded only on the results of observation and experiment, and were neither deduced from nor did they form part of any science. The fact then that various nations in the vicinity of Greece had reached a high state of civilisation does not justify us in assuming that they had studied mathematics. The only races with whom the Greeks of Asia Minor (amongst whom our history begins) were likely to have come into frequent contact were those inhabiting the eastern littoral of the Mediterranean; and Greek

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tradition uniformly assigned the special development of geometry to the Egyptians, and that of the science of numbers either to the Egyptians or to the Phoenicians. I discuss these subjects separately. First, as to the science of numbers. So far as the acquirements of the Phoenicians on this subject are concerned it is impossible to speak with certainty. The magnitude of the commercial transactions of Tyre and Sidon necessitated a considerable development of arithmetic, to which it is probable the name of science might be properly applied. A Babylonian table of the numerical value of the squares of a series of consecutive integers has been found, and this would seem to indicate that properties of numbers were studied. According to Strabo the Tyrians paid particular attention to the sciences of numbers, navigation, and astronomy; they had, we know, considerable commerce with their neighbours and kinsmen the Chaldaeans; and B¨ockh says that they regularly supplied the weights and measures used in Babylon. Now the Chaldaeans had certainly paid some attention to arithmetic and geometry, as is shown by their astronomical calculations; and, whatever was the extent of their attainments in arithmetic, it is almost certain that the Phoenicians were equally proficient, while it is likely that the knowledge of the latter, such as it was, was communicated to the Greeks. On the whole it seems probable that the early Greeks were largely indebted to the Phoenicians for their knowledge of practical arithmetic or the art of calculation, and perhaps also learnt from them a few properties of numbers. It may be worthy of note that Pythagoras was a Phoenician; and according to Herodotus, but this is more doubtful, Thales was also of that race. I may mention that the almost universal use of the abacus or swanpan rendered it easy for the ancients to add and subtract without any knowledge of theoretical arithmetic. These instruments will be described later in chapter vii; it will be sufficient here to say that they afford a concrete way of representing a number in the decimal scale, and enable the results of addition and subtraction to be obtained by a merely mechanical process. This, coupled with a means of representing the result in writing, was all that was required for practical purposes. We are able to speak with more certainty on the arithmetic of the Egyptians. About forty years ago a hieratic papyrus,1 forming part 1

See Ein mathematisches Handbuch der alten Aegypter, by A. Eisenlohr, second edition, Leipzig, 1891; see also Cantor, chap. i; and A Short History of Greek Mathematics, by J. Gow, Cambridge, 1884, arts. 12–14. Besides these authorities the

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of the Rhind collection in the British Museum, was deciphered, which has thrown considerable light on their mathematical attainments. The manuscript was written by a scribe named Ahmes at a date, according to Egyptologists, considerably more than a thousand years before Christ, and it is believed to be itself a copy, with emendations, of a treatise more than a thousand years older. The work is called “directions for knowing all dark things,” and consists of a collection of problems in arithmetic and geometry; the answers are given, but in general not the processes by which they are obtained. It appears to be a summary of rules and questions familiar to the priests. The first part deals with the reduction of fractions of the form 2/(2n + 1) to a sum of fractions each of whose numerators is unity: 1 1 1 1 2 is the sum of 24 , 58 , 174 , and 232 ; for example, Ahmes states that 29 1 1 1 2 and 97 is the sum of 56 , 679 , and 776 . In all the examples n is less than 50. Probably he had no rule for forming the component fractions, and the answers given represent the accumulated experiences of previous writers: in one solitary case, however, he has indicated his method, for, after having asserted that 23 is the sum of 12 and 16 , he adds that therefore two-thirds of one-fifth is equal to the sum of a half of a fifth 1 1 and a sixth of a fifth, that is, to 10 + 30 . That so much attention was paid to fractions is explained by the fact that in early times their treatment was found difficult. The Egyptians and Greeks simplified the problem by reducing a fraction to the sum of several fractions, in each of which the numerator was unity, the sole exception to this rule being the fraction 23 . This remained the Greek practice until the sixth century of our era. The Romans, on the other hand, generally kept the denominator constant and equal to twelve, expressing the fraction (approximately) as so many twelfths. The Babylonians did the same in astronomy, except that they used sixty as the constant denominator; and from them through the Greeks the modern division of a degree into sixty equal parts is derived. Thus in one way or the other the difficulty of having to consider changes in both numerator and denominator was evaded. To-day when using decimals we often keep a fixed denominator, thus reverting to the Roman practice. After considering fractions Ahmes proceeds to some examples of the fundamental processes of arithmetic. In multiplication he seems to have papyrus has been discussed in memoirs by L. Rodet, A. Favaro, V. Bobynin, and E. Weyr.

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relied on repeated additions. Thus in one numerical example, where he requires to multiply a certain number, say a, by 13, he first multiplies by 2 and gets 2a, then he doubles the results and gets 4a, then he again doubles the result and gets 8a, and lastly he adds together a, 4a, and 8a. Probably division was also performed by repeated subtractions, but, as he rarely explains the process by which he arrived at a result, this is not certain. After these examples Ahmes goes on to the solution of some simple numerical equations. For example, he says “heap, its seventh, its whole, it makes nineteen,” by which he means that the object is to find a number such that the sum of it and one-seventh of it shall be together equal to 19; and he gives as the answer 16 + 12 + 18 , which is correct. The arithmetical part of the papyrus indicates that he had some idea of algebraic symbols. The unknown quantity is always represented by the symbol which means a heap; addition is sometimes represented by a pair of legs walking forwards, subtraction by a pair of legs walking −. backwards or by a flight of arrows; and equality by the sign < The latter part of the book contains various geometrical problems to which I allude later. He concludes the work with some arithmeticoalgebraical questions, two of which deal with arithmetical progressions and seem to indicate that he knew how to sum such series. Second, as to the science of geometry. Geometry is supposed to have had its origin in land-surveying; but while it is difficult to say when the study of numbers and calculation—some knowledge of which is essential in any civilised state—became a science, it is comparatively easy to distinguish between the abstract reasonings of geometry and the practical rules of the land-surveyor. Some methods of land-surveying must have been practised from very early times, but the universal tradition of antiquity asserted that the origin of geometry was to be sought in Egypt. That it was not indigenous to Greece, and that it arose from the necessity of surveying, is rendered the more probable by the derivation of the word from γ˜ η, the earth, and μετρέω, I measure. Now the Greek geometricians, as far as we can judge by their extant works, always dealt with the science as an abstract one: they sought for theorems which should be absolutely true, and, at any rate in historical times, would have argued that to measure quantities in terms of a unit which might have been incommensurable with some of the magnitudes considered would have made their results mere approximations to the truth. The name does not therefore refer to their practice. It is not, however, unlikely that it indicates the use which was made of geome-

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try among the Egyptians from whom the Greeks learned it. This also agrees with the Greek traditions, which in themselves appear probable; for Herodotus states that the periodical inundations of the Nile (which swept away the landmarks in the valley of the river, and by altering its course increased or decreased the taxable value of the adjoining lands) rendered a tolerably accurate system of surveying indispensable, and thus led to a systematic study of the subject by the priests. We have no reason to think that any special attention was paid to geometry by the Phoenicians, or other neighbours of the Egyptians. A small piece of evidence which tends to show that the Jews had not paid much attention to it is to be found in the mistake made in their sacred books,1 where it is stated that the circumference of a circle is three times its diameter: the Babylonians2 also reckoned that π was equal to 3. Assuming, then, that a knowledge of geometry was first derived by the Greeks from Egypt, we must next discuss the range and nature of Egyptian geometry.3 That some geometrical results were known at a date anterior to Ahmes’s work seems clear if we admit, as we have reason to do, that, centuries before it was written, the following method of obtaining a right angle was used in laying out the groundplan of certain buildings. The Egyptians were very particular about the exact orientation of their temples; and they had therefore to obtain with accuracy a north and south line, as also an east and west line. By observing the points on the horizon where a star rose and set, and taking a plane midway between them, they could obtain a north and south line. To get an east and west line, which had to be drawn at right angles to this, certain professional “rope-fasteners” were employed. These men used a rope ABCD divided by knots or marks at B and C, so that the lengths AB, BC, CD were in the ratio 3 : 4 : 5. The length BC was placed along the north and south line, and pegs P and Q inserted at the knots B and C. The piece BA (keeping it stretched all the time) was then rotated round the peg P , and similarly the piece CD was rotated round the peg Q, until the ends A and D coincided; the point thus indicated was marked by a peg R. The result was to form a triangle P QR whose sides RP , P Q, QR were in the ratio 3 : 4 : 5. The angle of 1

I. Kings, chap. vii, verse 23, and II. Chronicles, chap. iv, verse 2. See J. Oppert, Journal Asiatique, August 1872, and October 1874. 3 See Eisenlohr; Cantor, chap. ii; Gow, arts. 75, 76; and Die Geometrie der alten Aegypter, by E. Weyr, Vienna, 1884. 2

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the triangle at P would then be a right angle, and the line P R would give an east and west line. A similar method is constantly used at the present time by practical engineers for measuring a right angle. The property employed can be deduced as a particular case of Euc. i, 48; and there is reason to think that the Egyptians were acquainted with the results of this proposition and of Euc. i, 47, for triangles whose sides are in the ratio mentioned above. They must also, there is little doubt, have known that the latter proposition was true for an isosceles right-angled triangle, as this is obvious if a floor be paved with tiles of that shape. But though these are interesting facts in the history of the Egyptian arts we must not press them too far as showing that geometry was then studied as a science. Our real knowledge of the nature of Egyptian geometry depends mainly on the Rhind papyrus. Ahmes commences that part of his papyrus which deals with geometry by giving some numerical instances of the contents of barns. Unluckily we do not know what was the usual shape of an Egyptian barn, but where it is defined by three linear measurements, say a, b, and c, the answer is always given as if he had formed the expression a × b × (c + 12 c). He next proceeds to find the areas of certain rectilineal figures; if the text be correctly interpreted, some of these results are wrong. He then goes on to find the area of a circular field of diameter 12—no unit of length being mentioned—and gives the result as (d − 19 d)2 , where d is the diameter of the circle: this is equivalent to taking 3.1604 as the value of π, the actual value being very approximately 3.1416. Lastly, Ahmes gives some problems on pyramids. These long proved incapable of interpretation, but Cantor and Eisenlohr have shown that Ahmes was attempting to find, by means of data obtained from the measurement of the external dimensions of a building, the ratio of certain other dimensions which could not be directly measured: his process is equivalent to determining the trigonometrical ratios of certain angles. The data and the results given agree closely with the dimensions of some of the existing pyramids. Perhaps all Ahmes’s geometrical results were intended only as approximations correct enough for practical purposes. It is noticeable that all the specimens of Egyptian geometry which we possess deal only with particular numerical problems and not with general theorems; and even if a result be stated as universally true, it was probably proved to be so only by a wide induction. We shall see later that Greek geometry was from its commencement deductive. There are reasons for thinking that Egyptian geometry and arithmetic

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made little or no progress subsequent to the date of Ahmes’s work; and though for nearly two hundred years after the time of Thales Egypt was recognised by the Greeks as an important school of mathematics, it would seem that, almost from the foundation of the Ionian school, the Greeks outstripped their former teachers. It may be added that Ahmes’s book gives us much that idea of Egyptian mathematics which we should have gathered from statements about it by various Greek and Latin authors, who lived centuries later. Previous to its translation it was commonly thought that these statements exaggerated the acquirements of the Egyptians, and its discovery must increase the weight to be attached to the testimony of these authorities. We know nothing of the applied mathematics (if there were any) of the Egyptians or Phoenicians. The astronomical attainments of the Egyptians and Chaldaeans were no doubt considerable, though they were chiefly the results of observation: the Phoenicians are said to have confined themselves to studying what was required for navigation. Astronomy, however, lies outside the range of this book. I do not like to conclude the chapter without a brief mention of the Chinese, since at one time it was asserted that they were familiar with the sciences of arithmetic, geometry, mechanics, optics, navigation, and astronomy nearly three thousand years ago, and a few writers were inclined to suspect (for no evidence was forthcoming) that some knowledge of this learning had filtered across Asia to the West. It is true that at a very early period the Chinese were acquainted with several geometrical or rather architectural implements, such as the rule, square, compasses, and level; with a few mechanical machines, such as the wheel and axle; that they knew of the characteristic property of the magnetic needle; and were aware that astronomical events occurred in cycles. But the careful investigations of L. A. S´edillot1 have shown that the Chinese made no serious attempt to classify or extend the few rules of arithmetic or geometry with which they were acquainted, or to explain the causes of the phenomena which they observed. The idea that the Chinese had made considerable progress in theoretical mathematics seems to have been due to a misapprehension of the Jesuit missionaries who went to China in the sixteenth century. 1

See Boncompagni’s Bulletino di bibliografia e di storia delle scienze matematiche e fisiche for May, 1868, vol. i, pp. 161–166. On Chinese mathematics, mostly of a later date, see Cantor, chap. xxxi.

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In the first place, they failed to distinguish between the original science of the Chinese and the views which they found prevalent on their arrival—the latter being founded on the work and teaching of Arab or Hindoo missionaries who had come to China in the course of the thirteenth century or later, and while there introduced a knowledge of spherical trigonometry. In the second place, finding that one of the most important government departments was known as the Board of Mathematics, they supposed that its function was to promote and superintend mathematical studies in the empire. Its duties were really confined to the annual preparation of an almanack, the dates and predictions in which regulated many affairs both in public and domestic life. All extant specimens of these almanacks are defective and, in many respects, inaccurate. The only geometrical theorem with which we can be certain that the ancient Chinese were acquainted is that in certain cases (namely, √ when the ratio of the sides is 3 : 4 : 5, or 1 : 1 : 2) the area of the square described on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares described on the sides. It is barely possible that a few geometrical theorems which can be demonstrated in the quasi-experimental way of superposition were also known to them. Their arithmetic was decimal in notation, but their knowledge seems to have been confined to the art of calculation by means of the swan-pan, and the power of expressing the results in writing. Our acquaintance with the early attainments of the Chinese, slight though it is, is more complete than in the case of most of their contemporaries. It is thus specially instructive, and serves to illustrate the fact that a nation may possess considerable skill in the applied arts while they are ignorant of the sciences on which those arts are founded. From the foregoing summary it will be seen that our knowledge of the mathematical attainments of those who preceded the Greeks is very limited; but we may reasonably infer that from one source or another the early Greeks learned the use of the abacus for practical calculations, symbols for recording the results, and as much mathematics as is contained or implied in the Rhind papyrus. It is probable that this sums up their indebtedness to other races. In the next six chapters I shall trace the development of mathematics under Greek influence.

9

FIRST PERIOD.

Mathematics under Greek Influence. This period begins with the teaching of Thales, circ. 600 b.c., and ends with the capture of Alexandria by the Mohammedans in or about 641 a.d. The characteristic feature of this period is the development of Geometry. It will be remembered that I commenced the last chapter by saying that the history of mathematics might be divided into three periods, namely, that of mathematics under Greek influence, that of the mathematics of the middle ages and of the renaissance, and lastly that of modern mathematics. The next four chapters (chapters ii, iii, iv and v) deal with the history of mathematics under Greek influence: to these it will be convenient to add one (chapter vi) on the Byzantine school, since through it the results of Greek mathematics were transmitted to western Europe; and another (chapter vii) on the systems of numeration which were ultimately displaced by the system introduced by the Arabs. I should add that many of the dates mentioned in these chapters are not known with certainty, and must be regarded as only approximately correct. There appeared in December 1921, just before this reprint was struck off, Sir T. L. Heath’s work in 2 volumes on the History of Greek Mathematics. This may now be taken as the standard authority for this period.

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CHAPTER II. the ionian and pythagorean schools.1 circ. 600 b.c.–400 b.c. With the foundation of the Ionian and Pythagorean schools we emerge from the region of antiquarian research and conjecture into the light of history. The materials at our disposal for estimating the knowledge of the philosophers of these schools previous to about the year 430 b.c. are, however, very scanty Not only have all but fragments of the different mathematical treatises then written been lost, but we possess no copy of the history of mathematics written about 325 b.c. by Eudemus (who was a pupil of Aristotle). Luckily Proclus, who about 450 a.d. wrote a commentary on the earlier part of Euclid’s Elements, was familiar with Eudemus’s work, and freely utilised it in his historical references. We have also a fragment of the General View of Mathematics written by Geminus about 50 b.c., in which the methods of proof used by the early Greek geometricians are compared with those current at a later date. In addition to these general statements we have biographies of a few of the leading mathematicians, and some scattered notes in various writers in which allusions are made to the lives and works of others. The original authorities are criticised and discussed at length in the works mentioned in the footnote to the heading of the chapter. 1

The history of these schools has been discussed by G. Loria in his Le Scienze Esatte nell’ Antica Grecia, Modena, 1893–1900; by Cantor, chaps. v–viii; by G. J. Allman in his Greek Geometry from Thales to Euclid, Dublin, 1889; by J. Gow, in his Greek Mathematics, Cambridge, 1884; by C. A. Bretschneider in his Die Geometrie und die Geometer vor Eukleides, Leipzig, 1870; and partially by H. Hankel in his posthumous Geschichte der Mathematik, Leipzig, 1874.

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The Ionian School. Thales.1 The founder of the earliest Greek school of mathematics and philosophy was Thales, one of the seven sages of Greece, who was born about 640 b.c. at Miletus, and died in the same town about 550 b.c. The materials for an account of his life consist of little more than a few anecdotes which have been handed down by tradition. During the early part of his life Thales was engaged partly in commerce and partly in public affairs; and to judge by two stories that have been preserved, he was then as distinguished for shrewdness in business and readiness in resource as he was subsequently celebrated in science. It is said that once when transporting some salt which was loaded on mules, one of the animals slipping in a stream got its load wet and so caused some of the salt to be dissolved, and finding its burden thus lightened it rolled over at the next ford to which it came; to break it of this trick Thales loaded it with rags and sponges which, by absorbing the water, made the load heavier and soon effectually cured it of its troublesome habit. At another time, according to Aristotle, when there was a prospect of an unusually abundant crop of olives Thales got possession of all the olive-presses of the district; and, having thus “cornered” them, he was able to make his own terms for lending them out, or buying the olives, and thus realized a large sum. These tales may be apocryphal, but it is certain that he must have had considerable reputation as a man of affairs and as a good engineer, since he was employed to construct an embankment so as to divert the river Halys in such a way as to permit of the construction of a ford. Probably it was as a merchant that Thales first went to Egypt, but during his leisure there he studied astronomy and geometry. He was middle-aged when he returned to Miletus; he seems then to have abandoned business and public life, and to have devoted himself to the study of philosophy and science—subjects which in the Ionian, Pythagorean, and perhaps also the Athenian schools, were closely connected: his views on philosophy do not here concern us. He continued to live at Miletus till his death circ. 550 b.c. We cannot form any exact idea as to how Thales presented his geometrical teaching. We infer, however, from Proclus that it consisted of a number of isolated propositions which were not arranged in a logical sequence, but that the proofs were deductive, so that the theorems were 1

See Loria, book I, chap. ii; Cantor, chap. v; Allman, chap. i.

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not a mere statement of an induction from a large number of special instances, as probably was the case with the Egyptian geometricians. The deductive character which he thus gave to the science is his chief claim to distinction. The following comprise the chief propositions that can now with reasonable probability be attributed to him; they are concerned with the geometry of angles and straight lines. (i) The angles at the base of an isosceles triangle are equal (Euc. i, 5). Proclus seems to imply that this was proved by taking another exactly equal isosceles triangle, turning it over, and then superposing it on the first—a sort of experimental demonstration. (ii) If two straight lines cut one another, the vertically opposite angles are equal (Euc. i, 15). Thales may have regarded this as obvious, for Proclus adds that Euclid was the first to give a strict proof of it. (iii) A triangle is determined if its base and base angles be given (cf. Euc. i, 26). Apparently this was applied to find the distance of a ship at sea—the base being a tower, and the base angles being obtained by observation. (iv) The sides of equiangular triangles are proportionals (Euc. vi, 4, or perhaps rather Euc. vi, 2). This is said to have been used by Thales when in Egypt to find the height of a pyramid. In a dialogue given by Plutarch, the speaker, addressing Thales, says, “Placing your stick at the end of the shadow of the pyramid, you made by the sun’s rays two triangles, and so proved that the [height of the] pyramid was to the [length of the] stick as the shadow of the pyramid to the shadow of the stick.” It would seem that the theorem was unknown to the Egyptians, and we are told that the king Amasis, who was present, was astonished at this application of abstract science. (v) A circle is bisected by any diameter. This may have been enunciated by Thales, but it must have been recognised as an obvious fact from the earliest times. (vi) The angle subtended by a diameter of a circle at any point in the circumference is a right angle (Euc. iii, 31). This appears to have been regarded as the most remarkable of the geometrical achievements of Thales, and it is stated that on inscribing a right-angled triangle in a circle he sacrificed an ox to the immortal gods. It has been conjectured that he may have come to this conclusion by noting that the diagonals of a rectangle are equal and bisect one another, and that therefore a rectangle can be inscribed in a circle. If so, and if he went on to apply proposition (i), he would have discovered that the sum of the angles of a

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right-angled triangle is equal to two right angles, a fact with which it is believed that he was acquainted. It has been remarked that the shape of the tiles used in paving floors may have suggested these results. On the whole it seems unlikely that he knew how to draw a perpendicular from a point to a line; but if he possessed this knowledge, it is possible he was also aware, as suggested by some modern commentators, that the sum of the angles of any triangle is equal to two right angles. As far as equilateral and right-angled triangles are concerned, we know from Eudemus that the first geometers proved the general property separately for three species of triangles, and it is not unlikely that they proved it thus. The area about a point can be filled by the angles of six equilateral triangles or tiles, hence the proposition is true for an equilateral triangle. Again, any two equal right-angled triangles can be placed in juxtaposition so as to form a rectangle, the sum of whose angles is four right angles; hence the proposition is true for a right-angled triangle. Lastly, any triangle can be split into the sum of two right-angled triangles by drawing a perpendicular from the biggest angle on the opposite side, and therefore again the proposition is true. The first of these proofs is evidently included in the last, but there is nothing improbable in the suggestion that the early Greek geometers continued to teach the first proposition in the form above given. Thales wrote on astronomy, and among his contemporaries was more famous as an astronomer than as a geometrician. A story runs that one night, when walking out, he was looking so intently at the stars that he tumbled into a ditch, on which an old woman exclaimed, “How can you tell what is going on in the sky when you can’t see what is lying at your own feet?”—an anecdote which was often quoted to illustrate the unpractical character of philosophers. Without going into astronomical details, it may be mentioned that he taught that a year contained about 365 days, and not (as is said to have been previously reckoned) twelve months of thirty days each. It is said that his predecessors occasionally intercalated a month to keep the seasons in their customary places, and if so they must have realized that the year contains, on the average, more than 360 days. There is some reason to think that he believed the earth to be a disc-like body floating on water. He predicted a solar eclipse which took place at or about the time he foretold; the actual date was either May 28, 585 b.c., or September 30, 609 b.c. But though this prophecy and its fulfilment gave extraordinary prestige to his teaching, and secured him the name of one of the seven sages of Greece, it is most likely that he only made

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use of one of the Egyptian or Chaldaean registers which stated that solar eclipses recur at intervals of about 18 years 11 days. Among the pupils of Thales were Anaximander, Anaximenes, Mamercus, and Mandryatus. Of the three mentioned last we know next to nothing. Anaximander was born in 611 b.c., and died in 545 b.c., and succeeded Thales as head of the school at Miletus. According to Suidas he wrote a treatise on geometry in which, tradition says, he paid particular attention to the properties of spheres, and dwelt at length on the philosophical ideas involved in the conception of infinity in space and time. He constructed terrestrial and celestial globes. Anaximander is alleged to have introduced the use of the style or gnomon into Greece. This, in principle, consisted only of a stick stuck upright in a horizontal piece of ground. It was originally used as a sun-dial, in which case it was placed at the centre of three concentric circles, so that every two hours the end of its shadow passed from one circle to another. Such sun-dials have been found at Pompeii and Tusculum. It is said that he employed these styles to determine his meridian (presumably by marking the lines of shadow cast by the style at sunrise and sunset on the same day, and taking the plane bisecting the angle so formed); and thence, by observing the time of year when the noon-altitude of the sun was greatest and least, he got the solstices; thence, by taking half the sum of the noon-altitudes of the sun at the two solstices, he found the inclination of the equator to the horizon (which determined the altitude of the place), and, by taking half their difference, he found the inclination of the ecliptic to the equator. There seems good reason to think that he did actually determine the latitude of Sparta, but it is more doubtful whether he really made the rest of these astronomical deductions. We need not here concern ourselves further with the successors of Thales. The school he established continued to flourish till about 400 b.c., but, as time went on, its members occupied themselves more and more with philosophy and less with mathematics. We know very little of the mathematicians comprised in it, but they would seem to have devoted most of their attention to astronomy. They exercised but slight influence on the further advance of Greek mathematics, which was made almost entirely under the influence of the Pythagoreans, who not only immensely developed the science of geometry, but created a science of numbers. If Thales was the first to direct general attention to geometry, it was Pythagoras, says Proclus, quoting from Eudemus, who

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“changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom and investigated its theorems in an . . . intellectual manner”; and it is accordingly to Pythagoras that we must now direct attention. The Pythagorean School. Pythagoras.1 Pythagoras was born at Samos about 569 b.c., perhaps of Tyrian parents, and died in 500 b.c. He was thus a contemporary of Thales. The details of his life are somewhat doubtful, but the following account is, I think, substantially correct. He studied first under Pherecydes of Syros, and then under Anaximander; by the latter he was recommended to go to Thebes, and there or at Memphis he spent some years. After leaving Egypt he travelled in Asia Minor, and then settled at Samos, where he gave lectures but without much success. About 529 b.c. he migrated to Sicily with his mother, and with a single disciple who seems to have been the sole fruit of his labours at Samos. Thence he went to Tarentum, but very shortly moved to Croton, a Dorian colony in the south of Italy. Here the schools that he opened were crowded with enthusiastic audiences; citizens of all ranks, especially those of the upper classes, attended, and even the women broke a law which forbade their going to public meetings and flocked to hear him. Amongst his most attentive auditors was Theano, the young and beautiful daughter of his host Milo, whom, in spite of the disparity of their ages, he married. She wrote a biography of her husband, but unfortunately it is lost. Pythagoras divided those who attended his lectures into two classes, whom we may term probationers and Pythagoreans. The majority were probationers, but it was only to the Pythagoreans that his chief discoveries were revealed. The latter formed a brotherhood with all things in common, holding the same philosophical and political beliefs, engaged in the same pursuits, and bound by oath not to reveal the teaching or secrets of the school; their food was simple; their discipline 1

See Loria, book I, chap. iii; Cantor, chaps. vi, vii; Allman, chap. ii; Hankel, pp. 92–111; Hoefer, Histoire des math´ematiques, Paris, third edition, 1886, pp. 87– 130; and various papers by S. P. Tannery. For an account of Pythagoras’s life, embodying the Pythagorean traditions, see the biography by Iamblichus, of which there are two or three English translations. Those who are interested in esoteric literature may like to see a modern attempt to reproduce the Pythagorean teaching in Pythagoras, by E. Schur´e, Eng. trans., London, 1906.

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severe; and their mode of life arranged to encourage self-command, temperance, purity, and obedience. This strict discipline and secret organisation gave the society a temporary supremacy in the state which brought on it the hatred of various classes; and, finally, instigated by his political opponents, the mob murdered Pythagoras and many of his followers. Though the political influence of the Pythagoreans was thus destroyed, they seem to have re-established themselves at once as a philosophical and mathematical society, with Tarentum as their headquarters, and they continued to flourish for more than a hundred years. Pythagoras himself did not publish any books; the assumption of his school was that all their knowledge was held in common and veiled from the outside world, and, further, that the glory of any fresh discovery must be referred back to their founder. Thus Hippasus (circ. 470 b.c.) is said to have been drowned for violating his oath by publicly boasting that he had added the dodecahedron to the number of regular solids enumerated by Pythagoras. Gradually, as the society became more scattered, this custom was abandoned, and treatises containing the substance of their teaching and doctrines were written. The first book of the kind was composed, about 370 b.c., by Philolaus, and we are told that Plato secured a copy of it. We may say that during the early part of the fifth century before Christ the Pythagoreans were considerably in advance of their contemporaries, but by the end of that time their more prominent discoveries and doctrines had become known to the outside world, and the centre of intellectual activity was transferred to Athens. Though it is impossible to separate precisely the discoveries of Pythagoras himself from those of his school of a later date, we know from Proclus that it was Pythagoras who gave geometry that rigorous character of deduction which it still bears, and made it the foundation of a liberal education; and there is reason to believe that he was the first to arrange the leading propositions of the subject in a logical order. It was also, according to Aristoxenus, the glory of his school that they raised arithmetic above the needs of merchants. It was their boast that they sought knowledge and not wealth, or in the language of one of their maxims, “a figure and a step forwards, not a figure to gain three oboli.” Pythagoras was primarily a moral reformer and philosopher, but his system of morality and philosophy was built on a mathematical foundation. His mathematical researches were, however, designed to lead

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up to a system of philosophy whose exposition was the main object of his teaching. The Pythagoreans began by dividing the mathematical subjects with which they dealt into four divisions: numbers absolute or arithmetic, numbers applied or music, magnitudes at rest or geometry, and magnitudes in motion or astronomy. This “quadrivium” was long considered as constituting the necessary and sufficient course of study for a liberal education. Even in the case of geometry and arithmetic (which are founded on inferences unconsciously made and common to all men) the Pythagorean presentation was involved with philosophy; and there is no doubt that their teaching of the sciences of astronomy, mechanics, and music (which can rest safely only on the results of conscious observation and experiment) was intermingled with metaphysics even more closely. It will be convenient to begin by describing their treatment of geometry and arithmetic. First, as to their geometry. Pythagoras probably knew and taught the substance of what is contained in the first two books of Euclid about parallels, triangles, and parallelograms, and was acquainted with a few other isolated theorems including some elementary propositions on irrational magnitudes; but it is suspected that many of his proofs were not rigorous, and in particular that the converse of a theorem was sometimes assumed without a proof. It is hardly necessary to say that we are unable to reproduce the whole body of Pythagorean teaching on this subject, but we gather from the notes of Proclus on Euclid, and from a few stray remarks in other writers, that it included the following propositions, most of which are on the geometry of areas. (i) It commenced with a number of definitions, which probably were rather statements connecting mathematical ideas with philosophy than explanations of the terms used. One has been preserved in the definition of a point as unity having position. (ii) The sum of the angles of a triangle was shown to be equal to two right angles (Euc. i, 32); and in the proof, which has been preserved, the results of the propositions Euc. i, 13 and the first part of Euc. i, 29 are quoted. The demonstration is substantially the same as that in Euclid, and it is most likely that the proofs there given of the two propositions last mentioned are also due to Pythagoras himself. (iii) Pythagoras certainly proved the properties of right-angled triangles which are given in Euc. i, 47 and i, 48. We know that the proofs of these propositions which are found in Euclid were of Euclid’s own invention; and a good deal of curiosity has been excited to discover what was the demonstration which was originally offered by Pythago-

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ras of the first of these theorems. It has been conjectured that not improbably it may have been one of the two following.1 A

F

E

B

K

G D

H

C

(α) Any square ABCD can be split up, as in Euc. ii, 4, into two squares BK and DK and two equal rectangles AK and CK: that is, it is equal to the square on F K, the square on EK, and four times the triangle AEF . But, if points be taken, G on BC, H on CD, and E on DA, so that BG, CH, and DE are each equal to AF , it can be easily shown that EF GH is a square, and that the triangles AEF , BF G, CGH, and DHE are equal: thus the square ABCD is also equal to the square on EF and four times the triangle AEF . Hence the square on EF is equal to the sum of the squares on F K and EK. A

B

D

C

(β) Let ABC be a right-angled triangle, A being the right angle. Draw AD perpendicular to BC. The triangles ABC and DBA are 1

A collection of a hundred proofs of Euc. i, 47 was published in the American Mathematical Monthly Journal, vols. iii. iv. v. vi. 1896–1899.

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similar,

Similarly Hence

∴ BC : AB = AB : BD. BC : AC = AC : DC. 2 AB + AC 2 = BC(BD + DC) = BC 2 .

This proof requires a knowledge of the results of Euc. ii, 2, vi, 4, and vi, 17, with all of which Pythagoras was acquainted. (iv) Pythagoras is credited by some writers with the discovery of the theorems Euc. i, 44, and i, 45, and with giving a solution of the problem Euc. ii, 14. It is said that on the discovery of the necessary construction for the problem last mentioned he sacrificed an ox, but as his school had all things in common the liberality was less striking than it seems at first. The Pythagoreans of a later date were aware of the extension given in Euc. vi, 25, and Allman thinks that Pythagoras himself was acquainted with it, but this must be regarded as doubtful. It will be noticed that Euc. ii, 14 provides a geometrical solution of the equation x2 = ab. (v) Pythagoras showed that the plane about a point could be completely filled by equilateral triangles, by squares, or by regular hexagons —results that must have been familiar wherever tiles of these shapes were in common use. (vi) The Pythagoreans were said to have attempted the quadrature of the circle: they stated that the circle was the most perfect of all plane figures. (vii) They knew that there were five regular solids inscribable in a sphere, which was itself, they said, the most perfect of all solids. (viii) From their phraseology in the science of numbers and from other occasional remarks, it would seem that they were acquainted with the methods used in the second and fifth books of Euclid, and knew something of irrational magnitudes. In particular, there is reason to believe that Pythagoras proved that the side and the diagonal of a square were incommensurable, and that it was this discovery which led the early Greeks to banish the conceptions of number and measurement from their geometry. A proof of this proposition which may be that due to Pythagoras is given below.1 1

See below, page 49.

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Next, as to their theory of numbers.1 In this Pythagoras was chiefly concerned with four different groups of problems which dealt respectively with polygonal numbers, with ratio and proportion, with the factors of numbers, and with numbers in series; but many of his arithmetical inquiries, and in particular the questions on polygonal numbers and proportion, were treated by geometrical methods. H

K

A

C

L

Pythagoras commenced his theory of arithmetic by dividing all numbers into even or odd: the odd numbers being termed gnomons. An odd number, such as 2n + 1, was regarded as the difference of two square numbers (n + 1)2 and n2 ; and the sum of the gnomons from 1 to 2n + 1 was stated to be a square number, viz. (n + 1)2 , its square root was termed a side. Products of two numbers were called plane, and if a product had no exact square root it was termed an oblong. A product of three numbers was called a solid number, and, if the three numbers were equal, a cube. All this has obvious reference to geometry, and the opinion is confirmed by Aristotle’s remark that when a gnomon is put round a square the figure remains a square though it is increased in dimensions. Thus, in the figure given above in which n is taken equal to 5, the gnomon AKC (containing 11 small squares) when put round the square AC (containing 52 small squares) makes a square HL (containing 62 small squares). It is possible that several 1

See the appendix Sur l’arithm´etique pythagorienne to S. P. Tannery’s La science hell`ene, Paris, 1887.

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of the numerical theorems due to Greek writers were discovered and proved by an analogous method: the abacus can be used for many of these demonstrations. The numbers (2n2 + 2n + 1), (2n2 + 2n), and (2n + 1) possessed special importance as representing the hypotenuse and two sides of a right-angled triangle: Cantor thinks that Pythagoras knew this fact before discovering the geometrical proposition Euc. i, 47. A more general expression for such numbers is (m2 +n2 ), 2mn, and (m2 −n2 ), or multiples of them: it will be noticed that the result obtained by Pythagoras can be deduced from these expressions by assuming m = n + 1; at a later time Archytas and Plato gave rules which are equivalent to taking n = 1; Diophantus knew the general expressions. After this preliminary discussion the Pythagoreans proceeded to the four special problems already alluded to. Pythagoras was himself acquainted with triangular numbers; polygonal numbers of a higher order were discussed by later members of the school. A triangular number represents the sum of a number of counters laid in rows on a plane; the bottom row containing n, and each succeeding row one less: it is therefore equal to the sum of the series n + (n − 1) + (n − 2) + . . . + 2 + 1, that is, to 21 n(n + 1). Thus the triangular number corresponding to 4 is 10. This is the explanation of the language of Pythagoras in the wellknown passage in Lucian where the merchant asks Pythagoras what he can teach him. Pythagoras replies “I will teach you how to count.” Merchant, “I know that already.” Pythagoras, “How do you count?” Merchant, “One, two, three, four—” Pythagoras, “Stop! what you take to be four is ten, a perfect triangle and our symbol.” The Pythagoreans are, on somewhat doubtful authority, said to have classified numbers by comparing them with the sum of their integral subdivisors or factors, calling a number excessive, perfect, or defective, according as the sum of these subdivisors was greater than, equal to, or less than the number: the classification at first being restricted to even numbers. The third group of problems which they considered dealt with numbers which formed a proportion; presumably these were discussed with the aid of geometry as is done in the fifth book of Euclid. Lastly, the Pythagoreans were concerned with series of numbers in arithmetical, geometrical, harmonical, and musical progressions. The three progressions first mentioned are well known; four integers are said to be in musical pro-

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gression when they are in the ratio a : 2ab/(a + b) : 12 (a + b) : b, for example, 6, 8, 9, and 12 are in musical progression. Of the Pythagorean treatment of the applied subjects of the quadrivium, and the philosophical theories founded on them, we know very little. It would seem that Pythagoras was much impressed by certain numerical relations which occur in nature. It has been suggested that he was acquainted with some of the simpler facts of crystallography. It is thought that he was aware that the notes sounded by a vibrating string depend on the length of the string, and in particular that lengths which gave a note, its fifth and its octave were in the ratio 2 : 3 : 4, forming terms in a musical progression. It would seem, too, that he believed that the distances of the astrological planets from the earth were also in musical progression, and that the heavenly bodies in their motion through space gave out harmonious sounds: hence the phrase the harmony of the spheres. These and similar conclusions seem to have suggested to him that the explanation of the order and harmony of the universe was to be found in the science of numbers, and that numbers are to some extent the cause of form as well as essential to its accurate measurement. He accordingly proceeded to attribute particular properties to particular numbers and geometrical figures. For example, he taught that the cause of colour was to be sought in properties of the number five, that the explanation of fire was to be discovered in the nature of the pyramid, and so on. I should not have alluded to this were it not that the Pythagorean tradition strengthened, or perhaps was chiefly responsible for the tendency of Greek writers to found the study of nature on philosophical conjectures and not on experimental observation—a tendency to which the defects of Hellenic science must be largely attributed. After the death of Pythagoras his teaching seems to have been carried on by Epicharmus and Hippasus, and subsequently by Philolaus (specially distinguished as an astronomer), Archippus, and Lysis. About a century after the murder of Pythagoras we find Archytas recognised as the head of the school. Archytas.1 Archytas, circ. 400 b.c., was one of the most influential citizens of Tarentum, and was made governor of the city no less 1

See Allman, chap. iv. A catalogue of the works of Archytas is given by Fabricius in his Bibliotheca Graeca, vol. i, p. 833: most of the fragments on philosophy were published by Thomas Gale in his Opuscula Mythologica, Cambridge, 1670; and by Thomas Taylor as an Appendix to his translation of Iamblichus’s Life of Pythagoras, London, 1818. See also the references given by Cantor, vol. i, p. 203.

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than seven times. His influence among his contemporaries was very great, and he used it with Dionysius on one occasion to save the life of Plato. He was noted for the attention he paid to the comfort and education of his slaves and of children in the city. He was drowned in a shipwreck near Tarentum, and his body washed on shore—a fit punishment, in the eyes of the more rigid Pythagoreans, for his having departed from the lines of study laid down by their founder. Several of the leaders of the Athenian school were among his pupils and friends, and it is believed that much of their work was due to his inspiration. The Pythagoreans at first made no attempt to apply their knowledge to mechanics, but Archytas is said to have treated it with the aid of geometry. He is alleged to have invented and worked out the theory of the pulley, and is credited with the construction of a flying bird and some other ingenious mechanical toys. He introduced various mechanical devices for constructing curves and solving problems. These were objected to by Plato, who thought that they destroyed the value of geometry as an intellectual exercise, and later Greek geometricians confined themselves to the use of two species of instruments, namely, rulers and compasses. Archytas was also interested in astronomy; he taught that the earth was a sphere rotating round its axis in twenty-four hours, and round which the heavenly bodies moved. Archytas was one of the first to give a solution of the problem to duplicate a cube, that is, to find the side of a cube whose volume is double that of a given cube. This was one of the most famous problems of antiquity.1 The construction given by Archytas is equivalent to the following. On the diameter OA of the base of a right circular cylinder describe a semicircle whose plane is perpendicular to the base of the cylinder. Let the plane containing this semicircle rotate round the generator through O, then the surface traced out by the semicircle will cut the cylinder in a tortuous curve. This curve will be cut by a right cone whose axis is OA and semivertical angle is (say) 60◦ in a point P , such that the projection of OP on the base of the cylinder will be to the radius of the cylinder in the ratio of the side of the required cube to that of the given cube. The proof given by Archytas is of course geometrical;2 it will be enough here to remark that in the course of it he shews himself acquainted with the results of the propositions Euc. iii, 18, Euc. iii, 35, and Euc. xi, 19. To shew analytically that the construction is correct, 1 2

See below, pp. 30, 34, 34. It is printed by Allman, pp. 111–113.

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take OA as the axis of x, and the generator through O as axis of z, then, with the usual notation in polar co-ordinates, and if a be the radius of the cylinder, we have for the equation of the surface described by the semicircle, r = 2a sin θ; for that of the cylinder, r sin θ = 2a cos φ; and for that of the cone, sin θ cos φ = 12 . These three surfaces cut in a point such that sin3 θ = 12 , and, therefore, if ρ be the projection of OP on the base of the cylinder, then ρ3 = (r sin θ)3 = 2a3 . Hence the volume of the cube whose side is ρ is twice that of a cube whose side is a. I mention the problem and give the construction used by Archytas to illustrate how considerable was the knowledge of the Pythagorean school at the time. Theodorus. Another Pythagorean of about the same date as Archytas was Theodorus of Cyrene, who is√said√to have √ proved √ √geomet√ 3, 5, 6, 7, 8, 10, rically that the numbers represented by √ √ √ √ √ √ 11, 12, 13, 14, 15, and 17 are incommensurable with unity. Theaetetus was one of his pupils. Perhaps Timaeus of Locri and Bryso of Heraclea should be mentioned as other distinguished Pythagoreans of this time. It is believed that Bryso attempted to find the area of a circle by inscribing and circumscribing squares, and finally obtained polygons between whose areas the area of the circle lay; but it is said that at some point he assumed that the area of the circle was the arithmetic mean between an inscribed and a circumscribed polygon. Other Greek Mathematical Schools in the Fifth Century b.c. It would be a mistake to suppose that Miletus and Tarentum were the only places where, in the fifth century, Greeks were engaged in laying a scientific foundation for the study of mathematics. These towns represented the centres of chief activity, but there were few cities or colonies of any importance where lectures on philosophy and geometry were not given. Among these smaller schools I may mention those at Chios, Elea, and Thrace. The best known philosopher of the School of Chios was Oenopides, who was born about 500 b.c., and died about 430 b.c. He devoted himself chiefly to astronomy, but he had studied geometry in Egypt, and is credited with the solution of two problems, namely, to draw a straight line from a given external point perpendicular to a given straight line (Euc. i, 12), and at a given point to construct an angle equal to a given angle (Euc. i, 23).

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Another important centre was at Elea in Italy. This was founded in Sicily by Xenophanes. He was followed by Parmenides, Zeno, and Melissus. The members of the Eleatic School were famous for the difficulties they raised in connection with questions that required the use of infinite series, such, for example, as the well-known paradox of Achilles and the tortoise, enunciated by Zeno, one of their most prominent members. Zeno was born in 495 b.c., and was executed at Elea in 435 b.c. in consequence of some conspiracy against the state; he was a pupil of Parmenides, with whom he visited Athens, circ. 455– 450 b.c. Zeno argued that if Achilles ran ten times as fast as a tortoise, yet if the tortoise had (say) 1000 yards start it could never be overtaken: for, when Achilles had gone the 1000 yards, the tortoise would still be 100 yards in front of him; by the time he had covered these 100 yards, it would still be 10 yards in front of him; and so on for ever: thus Achilles would get nearer and nearer to the tortoise, but never overtake it. The fallacy is usually explained by the argument that the time required to overtake the tortoise, can be divided into an infinite number of parts, as stated in the question, but these get smaller and smaller in geometrical progression, and the sum of them all is a finite time: after the lapse of that time Achilles would be in front of the tortoise. Probably Zeno would have replied that this argument rests on the assumption that space is infinitely divisible, which is the question under discussion: he himself asserted that magnitudes are not infinitely divisible. These paradoxes made the Greeks look with suspicion on the use of infinitesimals, and ultimately led to the invention of the method of exhaustions. The Atomistic School, having its headquarters in Thrace, was another important centre. This was founded by Leucippus, who was a pupil of Zeno. He was succeeded by Democritus and Epicurus. Its most famous mathematician was Democritus, born at Abdera in 460 b.c., and said to have died in 370 b.c., who, besides philosophical works, wrote on plane and solid geometry, incommensurable lines, perspective, and numbers. These works are all lost. From the Archimedean MS., discovered by Heiberg in 1906, it would seem that Democritus enunciated, but without a proof, the proposition that the volume of a pyramid is equal to one-third that of a prism of an equal base and of equal height. But though several distinguished individual philosophers may be mentioned who, during the fifth century, lectured at different cities,

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they mostly seem to have drawn their inspiration from Tarentum, and towards the end of the century to have looked to Athens as the intellectual capital of the Greek world; and it is to the Athenian schools that we owe the next great advance in mathematics.

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CHAPTER III. the schools of athens and cyzicus.1 circ. 420 b.c.–300 b.c. It was towards the close of the fifth century before Christ that Athens first became the chief centre of mathematical studies. Several causes conspired to bring this about. During that century she had become, partly by commerce, partly by appropriating for her own purposes the contributions of her allies, the most wealthy city in Greece; and the genius of her statesmen had made her the centre on which the politics of the peninsula turned. Moreover, whatever states disputed her claim to political supremacy her intellectual pre-eminence was admitted by all. There was no school of thought which had not at some time in that century been represented at Athens by one or more of its leading thinkers; and the ideas of the new science, which was being so eagerly studied in Asia Minor and Graecia Magna, had been brought before the Athenians on various occasions. Anaxagoras. Amongst the most important of the philosophers who resided at Athens and prepared the way for the Athenian school I may mention Anaxagoras of Clazomenae, who was almost the last philosopher of the Ionian school. He was born in 500 b.c., and died in 428 b.c. He seems to have settled at Athens about 440 b.c., and there taught the results of the Ionian philosophy. Like all members of that school he was much interested in astronomy. He asserted that 1

The history of these schools is discussed at length in G. Loria’s Le Scienze Esatte nell’ Antica Grecia, Modena, 1893–1900; in G. J. Allman’s Greek Geometry from Thales to Euclid, Dublin, 1889; and in J. Gow’s Greek Mathematics, Cambridge, 1884; it is also treated by Cantor, chaps. ix, x, and xi; by Hankel, pp. 111–156; and by C. A. Bretschneider in his Die Geometrie und die Geometer vor Eukleides, Leipzig, 1870; a critical account of the original authorities is given by S. P. Tannery in his G´eom´etrie Grecque, Paris, 1887, and other papers.

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the sun was larger than the Peloponnesus: this opinion, together with some attempts he had made to explain various physical phenomena which had been previously supposed to be due to the direct action of the gods, led to a prosecution for impiety, and he was convicted. While in prison he is said to have written a treatise on the quadrature of the circle. The Sophists. The sophists can hardly be considered as belonging to the Athenian school, any more than Anaxagoras can; but like him they immediately preceded and prepared the way for it, so that it is desirable to devote a few words to them. One condition for success in public life at Athens was the power of speaking well, and as the wealth and power of the city increased a considerable number of “sophists” settled there who undertook amongst other things to teach the art of oratory. Many of them also directed the general education of their pupils, of which geometry usually formed a part. We are told that two of those who are usually termed sophists made a special study of geometry—these were Hippias of Elis and Antipho, and one made a special study of astronomy—this was Meton, after whom the metonic cycle is named. Hippias. The first of these geometricians, Hippias of Elis (circ. 420 b.c.), is described as an expert arithmetician, but he is best known to us through his invention of a curve called the quadratrix, by means of which an angle can be trisected, or indeed divided in any given ratio. If the radius of a circle rotate uniformly round the centre O from the position OA through a right angle to OB, and in the same time a straight line drawn perpendicular to OB move uniformly parallel to itself from the position OA to BC, the locus of their intersection will be the quadratrix. Let OR and M Q be the position of these lines at any time; and let them cut in P , a point on the curve. Then angle AOP : angle AOB = OM : OB. Similarly, if OR0 be another position of the radius, angle AOP 0 : angle AOB = OM 0 : OB ∴ angle AOP : angle AOP 0 = OM : OM 0 ; ∴ angle AOP 0 : angle P 0 OP = OM 0 : M M. Hence, if the angle AOP be given, and it be required to divide it in any given ratio, it is sufficient to divide OM in that ratio at M 0 and draw the line M 0 P 0 ; then OP 0 will divide AOP in the required ratio.

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C R

M

M0

Q

P

R0 P0

O

A

If OA be taken as the initial line, OP = r, the angle AOP = θ, and OA = a, we have θ : 12 π = r sin θ : a, and the equation of the curve is πr = 2aθ cosec θ. Hippias devised an instrument to construct the curve mechanically; but constructions which involved the use of any mathematical instruments except a ruler and a pair of compasses were objected to by Plato, and rejected by most geometricians of a subsequent date. Antipho. The second sophist whom I mentioned was Antipho (circ. 420 b.c.). He is one of the very few writers among the ancients who attempted to find the area of a circle by considering it as the limit of an inscribed regular polygon with an infinite number of sides. He began by inscribing an equilateral triangle (or, according to some accounts, a square); on each side he inscribed in the smaller segment an isosceles triangle, and so on ad infinitum. This method of attacking the quadrature problem is similar to that described above as used by Bryso of Heraclea. No doubt there were other cities in Greece besides Athens where similar and equally meritorious work was being done, though the record of it has now been lost; I have mentioned here the investigations of these three writers, chiefly because they were the immediate predecessors of those who created the Athenian school. The Schools of Athens and Cyzicus. The history of the Athenian school begins with the teaching of Hippocrates about 420 b.c.; the school was established on a permanent basis by the labours

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of Plato and Eudoxus; and, together with the neighbouring school of Cyzicus, continued to extend on the lines laid down by these three geometricians until the foundation (about 300 b.c.) of the university at Alexandria drew thither most of the talent of Greece. Eudoxus, who was amongst the most distinguished of the Athenian mathematicians, is also reckoned as the founder of the school at Cyzicus. The connection between this school and that of Athens was very close, and it is now impossible to disentangle their histories. It is said that Hippocrates, Plato, and Theaetetus belonged to the Athenian school; while Eudoxus, Menaechmus, and Aristaeus belonged to that of Cyzicus. There was always a constant intercourse between the two schools, the earliest members of both had been under the influence either of Archytas or of his pupil Theodorus of Cyrene, and there was no difference in their treatment of the subject, so that they may be conveniently treated together. Before discussing the work of the geometricians of these schools in detail I may note that they were especially interested in three problems:1 namely (i), the duplication of a cube, that is, the determination of the side of a cube whose volume is double that of a given cube; (ii) the trisection of an angle; and (iii) the squaring of a circle, that is, the determination of a square whose area is equal to that of a given circle. Now the first two of these problems (considered analytically) require the solution of a cubic equation; and, since a construction by means of circles (whose equations are of the form x2 + y 2 + ax + by + c = 0) and straight lines (whose equations are of the form x + βy + γ = 0) cannot be equivalent to the solution of a cubic equation, the problems are insoluble if in our constructions we restrict ourselves to the use of circles and straight lines, that is, to Euclidean geometry. If the use of the conic sections be permitted, both of these questions can be solved in many ways. The third problem is equivalent to finding a rectangle whose sides are equal respectively to the radius and to the semiperimeter of the circle. These lines have been long known to be incommensurable, but it is only recently that it has been shewn by Lindemann that their ratio cannot be the root of a rational algebraical equation. Hence this problem also is insoluble by Euclidean geometry. The Athenians and Cyzicians were thus destined to fail in all three 1

On these problems, solutions of them, and the authorities for their history, see my Mathematical Recreations and Problems, London, ninth edition, 1920, chap. xiv.

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problems, but the attempts to solve them led to the discovery of many new theorems and processes. Besides attacking these problems the later Platonic school collected all the geometrical theorems then known and arranged them systematically. These collections comprised the bulk of the propositions in Euclid’s Elements, books i–ix, xi, and xii, together with some of the more elementary theorems in conic sections. Hippocrates. Hippocrates of Chios (who must be carefully distinguished from his contemporary, Hippocrates of Cos, the celebrated physician) was one of the greatest of the Greek geometricians. He was born about 470 b.c. at Chios, and began life as a merchant. The accounts differ as to whether he was swindled by the Athenian customhouse officials who were stationed at the Chersonese, or whether one of his vessels was captured by an Athenian pirate near Byzantium; but at any rate somewhere about 430 b.c. he came to Athens to try to recover his property in the law courts. A foreigner was not likely to succeed in such a case, and the Athenians seem only to have laughed at him for his simplicity, first in allowing himself to be cheated, and then in hoping to recover his money. While prosecuting his cause he attended the lectures of various philosophers, and finally (in all probability to earn a livelihood) opened a school of geometry himself. He seems to have been well acquainted with the Pythagorean philosophy, though there is no sufficient authority for the statement that he was ever initiated as a Pythagorean. He wrote the first elementary text-book of geometry, a text-book on which probably Euclid’s Elements was founded; and therefore he may be said to have sketched out the lines on which geometry is still taught in English schools. It is supposed that the use of letters in diagrams to describe a figure was made by him or introduced about this time, as he employs expressions such as “the point on which the letter A stands” and “the line on which AB is marked.” Cantor, however, thinks that the Pythagoreans had previously been accustomed to represent the five vertices of the pentagram-star by the letters υ γ ι θ α; and though this was a single instance, perhaps they may have used the method generally. The Indian geometers never employed letters to aid them in the description of their figures. Hippocrates also denoted the square on a line by the word δύναμις, and thus gave the technical meaning to the word power which it still retains in algebra: there is reason to think that this use of the word was derived from the Pythagoreans, who are said to have enunciated the result of the proposition Euc. i, 47, in the

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form that “the total power of the sides of a right-angled triangle is the same as that of the hypotenuse.” In this text-book Hippocrates introduced the method of “reducing” one theorem to another, which being proved, the thing proposed necessarily follows; of this method the reductio ad absurdum is an illustration. No doubt the principle had been used occasionally before, but he drew attention to it as a legitimate mode of proof which was capable of numerous applications. He elaborated the geometry of the circle: proving, among other propositions, that similar segments of a circle contain equal angles; that the angle subtended by the chord of a circle is greater than, equal to, or less than a right angle as the segment of the circle containing it is less than, equal to, or greater than a semicircle (Euc. iii, 31); and probably several other of the propositions in the third book of Euclid. It is most likely that he also established the propositions that [similar] circles are to one another as the squares of their diameters (Euc. xii, 2), and that similar segments are as the squares of their chords. The proof given in Euclid of the first of these theorems is believed to be due to Hippocrates. The most celebrated discoveries of Hippocrates were, however, in connection with the quadrature of the circle and the duplication of the cube, and owing to his influence these problems played a prominent part in the history of the Athenian school. The following propositions will sufficiently illustrate the method by which he attacked the quadrature problem. A

F G

B

D E

O

C

(α) He commenced by finding the area of a lune contained between a semicircle and a quadrantal arc standing on the same chord. This he did as follows. Let ABC be an isosceles right-angled triangle inscribed in the semicircle ABOC, whose centre is O. On AB and AC as diameters

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describe semicircles as in the figure. Then, since by Euc. i, 47, sq. on BC = sq. on AC + sq. on AB, therefore, by Euc. xii, 2, area

1 2

on BC = area

1 2

on AC + area

1 2

on AB.

Take away the common parts ∴ area 4ABC = sum of areas of lunes AECD and AF BG. Hence the area of the lune AECD is equal to half that of the triangle ABC. B

C

E F

D

O

A

(β) He next inscribed half a regular hexagon ABCD in a semicircle whose centre was O, and on OA, AB, BC, and CD as diameters described semicircles of which those on OA and AB are drawn in the figure. Then AD is double any of the lines OA, AB, BC, and CD, ∴ sq. on AD = sum of sqs. on OA, AB, BC, and CD, ∴ area ABCD = sum of areas of 12 s on OA, AB, BC, and CD. 1 2

Take away the common parts ∴ area trapezium ABCD = 3 lune AEBF + 21 on OA.

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If therefore the area of this latter lune be known, so is that of the semicircle described on OA as diameter. According to Simplicius, Hippocrates assumed that the area of this lune was the same as the area of the lune found in proposition (α); if this be so, he was of course mistaken, as in this case he is dealing with a lune contained between a semicircle and a sextantal arc standing on the same chord; but it seems more probable that Simplicius misunderstood Hippocrates. Hippocrates also enunciated various other theorems connected with lunes (which have been collected by Bretschneider and by Allman) of which the theorem last given is a typical example. I believe that they are the earliest instances in which areas bounded by curves were determined by geometry. The other problem to which Hippocrates turned his attention was the duplication of a cube, that is, the determination of the side of a cube whose volume is double that of a given cube. This problem was known in ancient times as the Delian problem, in consequence of a legend that the Delians had consulted Plato on the subject. In one form of the story, which is related by Philoponus, it is asserted that the Athenians in 430 b.c., when suffering from the plague of eruptive typhoid fever, consulted the oracle at Delos as to how they could stop it. Apollo replied that they must double the size of his altar which was in the form of a cube. To the unlearned suppliants nothing seemed more easy, and a new altar was constructed either having each of its edges double that of the old one (from which it followed that the volume was increased eightfold) or by placing a similar cubic altar next to the old one. Whereupon, according to the legend, the indignant god made the pestilence worse than before, and informed a fresh deputation that it was useless to trifle with him, as his new altar must be a cube and have a volume exactly double that of his old one. Suspecting a mystery the Athenians applied to Plato, who referred them to the geometricians, and especially to Euclid, who had made a special study of the problem. The introduction of the names of Plato and Euclid is an obvious anachronism. Eratosthenes gives a somewhat similar account of its origin, but with king Minos as the propounder of the problem. Hippocrates reduced the problem of duplicating the cube to that of finding two means between one straight line (a), and another twice as long (2a). If these means be x and y, we have a : x = x : y = y : 2a, from which it follows that x3 = 2a3 . It is in this form that the problem is usually presented now. Hippocrates did not succeed in finding a construction for these means.

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Plato. The next philosopher of the Athenian school who requires mention here was Plato. He was born at Athens in 429 b.c., and was, as is well known, a pupil for eight years of Socrates; much of the teaching of the latter is inferred from Plato’s dialogues. After the execution of his master in 399 b.c. Plato left Athens, and being possessed of considerable wealth he spent some years in travelling; it was during this time that he studied mathematics. He visited Egypt with Eudoxus, and Strabo says that in his time the apartments they occupied at Heliopolis were still shewn. Thence Plato went to Cyrene, where he studied under Theodorus. Next he moved to Italy, where he became intimate with Archytas the then head of the Pythagorean school, Eurytas of Metapontum, and Timaeus of Locri. He returned to Athens about the year 380 b.c., and formed a school of students in a suburban gymnasium called the “Academy.” He died in 348 b.c. Plato, like Pythagoras, was primarily a philosopher, and perhaps his philosophy should be regarded as founded on the Pythagorean rather than on the Socratic teaching. At any rate it, like that of the Pythagoreans, was coloured with the idea that the secret of the universe is to be found in number and in form; hence, as Eudemus says, “he exhibited on every occasion the remarkable connection between mathematics and philosophy.” All the authorities agree that, unlike many later philosophers, he made a study of geometry or some exact science an indispensable preliminary to that of philosophy. The inscription over the entrance to his school ran “Let none ignorant of geometry enter my door,” and on one occasion an applicant who knew no geometry is said to have been refused admission as a student. Plato’s position as one of the masters of the Athenian mathematical school rests not so much on his individual discoveries and writings as on the extraordinary influence he exerted on his contemporaries and successors. Thus the objection that he expressed to the use in the construction of curves of any instruments other than rulers and compasses was at once accepted as a canon which must be observed in such problems. It is probably due to Plato that subsequent geometricians began the subject with a carefully compiled series of definitions, postulates, and axioms. He also systematized the methods which could be used in attacking mathematical questions, and in particular directed attention to the value of analysis. The analytical method of proof begins by assuming that the theorem or problem is solved, and thence deducing some result: if the result be false, the theorem is not true or the problem is incapable of solution: if the result be true, and if the

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steps be reversible, we get (by reversing them) a synthetic proof; but if the steps be not reversible, no conclusion can be drawn. Numerous illustrations of the method will be found in any modern text-book on geometry. If the classification of the methods of legitimate induction given by Mill in his work on logic had been universally accepted and every new discovery in science had been justified by a reference to the rules there laid down, he would, I imagine, have occupied a position in reference to modern science somewhat analogous to that which Plato occupied in regard to the mathematics of his time. The following is the only extant theorem traditionally attributed to Plato. If CAB and DAB be two right-angled triangles, having one side, AB, common, their other sides, AD and BC, parallel, and their hypotenuses, AC and BD, at right angles, then, if these hypotenuses cut in P , we have P C : P B = P B : P A = P A : P D. This theorem was used in duplicating the cube, for, if such triangles can be constructed having P D = 2P C, the problem will be solved. It is easy to make an instrument by which the triangles can be constructed. Eudoxus.1 Of Eudoxus, the third great mathematician of the Athenian school and the founder of that at Cyzicus, we know very little. He was born in Cnidus in 408 b.c. Like Plato, he went to Tarentum and studied under Archytas the then head of the Pythagoreans. Subsequently he travelled with Plato to Egypt, and then settled at Cyzicus, where he founded the school of that name. Finally he and his pupils moved to Athens. There he seems to have taken some part in public affairs, and to have practised medicine; but the hostility of Plato and his own unpopularity as a foreigner made his position uncomfortable, and he returned to Cyzicus or Cnidus shortly before his death. He died while on a journey to Egypt in 355 b.c. His mathematical work seems to have been of a high order of excellence. He discovered most of what we now know as the fifth book of Euclid, and proved it in much the same form as that in which it is there given. He discovered some theorems on what was called “the golden section.” A H B The problem to cut a line AB in the golden section, that is, to divide it, say at H, in extreme and mean ratio 1

The works of Eudoxus were discussed in considerable detail by H. K¨ unssberg of Dinkelsb¨ uhl in 1888 and 1890; see also the authorities mentioned above in the footnote on p. 27.

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(that is, so that AB : AH = AH : HB) is solved in Euc. ii, 11, and probably was known to the Pythagoreans at an early date. If we denote AB by l, AH by a, and HB by b, the theorems that Eudoxus proved are equivalent to the following algebraical identities. (i) (a + 21 l)2 = 5( 12 l)2 . (ii) Conversely, if (i) be true, and AH be taken equal to a, then AB will be divided at H in a golden section. (iii) (b + 21 a)2 = 5( 12 a2 ). (iv) l2 + b2 = 3a2 . (v) l + a : l = l : a, which gives another golden section. These propositions were subsequently put by Euclid as the first five propositions of his thirteenth book, but they might have been equally well placed towards the end of the second book. All of them are obvious algebraically, since l = a + b and a2 = bl. Eudoxus further established the “method of exhaustions”; which depends on the proposition that “if from the greater of two unequal magnitudes there be taken more than its half, and from the remainder more than its half, and so on, there will at length remain a magnitude less than the least of the proposed magnitudes.” This proposition was placed by Euclid as the first proposition of the tenth book of his Elements, but in most modern school editions it is printed at the beginning of the twelfth book. By the aid of this theorem the ancient geometers were able to avoid the use of infinitesimals: the method is rigorous, but awkward of application. A good illustration of its use is to be found in the demonstration of Euc. xii, 2, namely, that the square of the radius of one circle is to the square of the radius of another circle as the area of the first circle is to an area which is neither less nor greater than the area of the second circle, and which therefore must be exactly equal to it: the proof given by Euclid is (as was usual) completed by a reductio ad absurdum. Eudoxus applied the principle to shew that the volume of a pyramid or a cone is one-third that of the prism or the cylinder on the same base and of the same altitude (Euc. xii, 7 and 10). It is believed that he proved that the volumes of two spheres were to one another as the cubes of their radii; some writers attribute the proposition Euc. xii, 2 to him, and not to Hippocrates. Eudoxus also considered certain curves other than the circle. There is no authority for the statement made in some old books that these were conic sections, and recent investigations have shewn that the assertion (which I repeated in the earlier editions of this book) that they were plane sections of the anchor-ring is also improbable. It seems most likely that they were tortuous curves; whatever they were, he applied them in explaining the apparent motions of the planets as seen from the earth.

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Eudoxus constructed an orrery, and wrote a treatise on practical astronomy, in which he supposed a number of moving spheres to which the sun, moon, and stars were attached, and which by their rotation produced the effects observed. In all he required twenty-seven spheres. As observations became more accurate, subsequent astronomers who accepted the theory had continually to introduce fresh spheres to make the theory agree with the facts. The work of Aratus on astronomy, which was written about 300 b.c. and is still extant, is founded on that of Eudoxus. Plato and Eudoxus were contemporaries. Among Plato’s pupils were the mathematicians Leodamas, Neocleides, Amyclas, and to their school also belonged Leon, Theudius (both of whom wrote text-books on plane geometry), Cyzicenus, Thasus, Hermotimus, Philippus, and Theaetetus. Among the pupils of Eudoxus are reckoned Menaechmus, his brother Dinostratus (who applied the quadratrix to the duplication and trisection problems), and Aristaeus. Menaechmus. Of the above-mentioned mathematicians Menaechmus requires special mention. He was born about 375 b.c., and died about 325 b.c. Probably he succeeded Eudoxus as head of the school at Cyzicus, where he acquired great reputation as a teacher of geometry, and was for that reason appointed one of the tutors of Alexander the Great. In answer to his pupil’s request to make his proofs shorter, Menaechmus made the well-known reply that though in the country there are private and even royal roads, yet in geometry there is only one road for all. Menaechmus was the first to discuss the conic sections, which were long called the Menaechmian triads. He divided them into three classes, and investigated their properties, not by taking different plane sections of a fixed cone, but by keeping his plane fixed and cutting it by different cones. He shewed that the section of a right cone by a plane perpendicular to a generator is an ellipse, if the cone be acute-angled; a parabola, if it be right-angled; and a hyperbola, if it be obtuse-angled; and he gave a mechanical construction for curves of each class. It seems almost certain that he was acquainted with the fundamental properties of these curves; but some writers think that he failed to connect them with the sections of the cone which he had discovered, and there is no doubt that he regarded the latter not as plane loci but as curves drawn on the surface of a cone. He also shewed how these curves could be used in either of the two following ways to give a solution of the problem to duplicate a cube. In

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the first of these, he pointed out that two parabolas having a common vertex, axes at right angles, and such that the latus rectum of the one is double that of the other will intersect in another point whose abscissa (or ordinate) will give a solution; for (using analysis) if the equations of the parabolas be y 2 = 2ax and x2 = ay, they intersect in a point whose abscissa is given by x3 = 2a3 . It is probable that this method was suggested by the form in which Hippocrates had cast the problem; namely, to find x and y so that a : x = x : y = y : 2a, whence we have x2 = ay and y 2 = 2ax. The second solution given by Menaechmus was as follows. Describe a parabola of latus rectum l. Next describe a rectangular hyperbola, the length of whose real axis is 4l, and having for its asymptotes the tangent at the vertex of the parabola and the axis of the parabola. Then the ordinate and the abscissa of the point of intersection of these curves are the mean proportionals between l and 2l. This is at once obvious by analysis. The curves are x2 = ly and xy = 2l2 . These cut in a point determined by x3 = 2l3 and y 3 = 4l3 . Hence l : x = x : y = y : 2l. Aristaeus and Theaetetus. Of the other members of these schools, Aristaeus and Theaetetus, whose works are entirely lost, were mathematicians of repute. We know that Aristaeus wrote on the five regular solids and on conic sections, and that Theaetetus developed the theory of incommensurable magnitudes. The only theorem we can now definitely ascribe to the latter is that given by Euclid in the ninth proposition of the tenth book of the Elements, namely, that the squares on two commensurable right lines have one to the other a ratio which a square number has to a square number (and conversely); but the squares on two incommensurable right lines have one to the other a ratio which cannot be expressed as that of a square number to a square number (and conversely). This theorem includes the results given by Theodorus.1 The contemporaries or successors of these mathematicians wrote some fresh text-books on the elements of geometry and the conic sections, introduced problems concerned with finding loci, and systematized the knowledge already acquired, but they originated no new methods of research. Aristotle. An account of the Athenian school would be incomplete if there were no mention of Aristotle, who was born at Stagira in Macedonia in 384 b.c. and died at Chalcis in Euboea in 322 b.c. Aris1

See above, p. 24.

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totle, however, deeply interested though he was in natural philosophy, was chiefly concerned with mathematics and mathematical physics as supplying illustrations of correct reasoning. A small book containing a few questions on mechanics which is sometimes attributed to him is of doubtful authority; but, though in all probability it is not his work, it is interesting, partly as shewing that the principles of mechanics were beginning to excite attention, and partly as containing the earliest known employment of letters to indicate magnitudes. The most instructive parts of the book are the dynamical proof of the parallelogram of forces for the direction of the resultant, and the statement, in effect, that if α be a force, β the mass to which it is applied, γ the distance through which it is moved, and δ the time of the motion, then α will move 12 β through 2γ in the time δ, or through γ in the time 12 δ: but the author goes on to say that it does not follow that 21 α will move β through 12 γ in the time δ, because 12 α may not be able to move β at all; for 100 men may drag a ship 100 yards, but it does not follow that one man can drag it one yard. The first part of this statement is correct and is equivalent to the statement that an impulse is proportional to the momentum produced, but the second part is wrong. The author also states the fact that what is gained in power is lost in speed, and therefore that two weights which keep a [weightless] lever in equilibrium are inversely proportional to the arms of the lever; this, he says, is the explanation why it is easier to extract teeth with a pair of pincers than with the fingers. Among other questions raised, but not answered, are why a projectile should ever stop, and why carriages with large wheels are easier to move than those with small. I ought to add that the book contains some gross blunders, and as a whole is not as able or suggestive as might be inferred from the above extracts. In fact, here as elsewhere, the Greeks did not sufficiently realise that the fundamental facts on which the mathematical treatment of mechanics must be based can be established only by carefully devised observations and experiments.

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CHAPTER IV. the first alexandrian school.1 circ. 300 b.c.–30 b.c. The earliest attempt to found a university, as we understand the word, was made at Alexandria. Richly endowed, supplied with lecture rooms, libraries, museums, laboratories, and gardens, it became at once the intellectual metropolis of the Greek race, and remained so for a thousand years. It was particularly fortunate in producing within the first century of its existence three of the greatest mathematicians of antiquity—Euclid, Archimedes, and Apollonius. They laid down the lines on which mathematics subsequently developed, and treated it as a subject distinct from philosophy: hence the foundation of the Alexandrian Schools is rightly taken as the commencement of a new era. Thenceforward, until the destruction of the city by the Arabs in 641 a.d., the history of mathematics centres more or less round that of Alexandria; for this reason the Alexandrian Schools are commonly taken to include all Greek mathematicians of their time. The city and university of Alexandria were created under the following circumstances. Alexander the Great had ascended the throne of Macedonia in 336 b.c. at the early age of twenty, and by 332 b.c. he had conquered or subdued Greece, Asia Minor, and Egypt. Following 1

The history of the Alexandrian Schools is discussed by G. Loria in his Le Scienze Esatte nell’ Antica Grecia, Modena, 1893–1900; by Cantor, chaps. xii–xxiii; and by J. Gow in his History of Greek Mathematics, Cambridge, 1884. The subject of Greek algebra is treated by E. H. F. Nesselmann in his Die Algebra der Griechen, Berlin, 1842; see also L. Matthiessen, Grundz¨ uge der antiken und modernen Algebra der litteralen Gleichungen, Leipzig, 1878. The Greek treatment of the conic sections forms the subject of Die Lehre von den Kegelschnitten in Altertum, by H. G. Zeuthen, Copenhagen, 1886. The materials for the history of these schools have been subjected to a searching criticism by S. P. Tannery, and most of his papers are collected in his G´eom´etrie Grecque, Paris, 1887.

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the plan he adopted whenever a commanding site had been left unoccupied, he founded a new city on the Mediterranean near one mouth of the Nile; and he himself sketched out the ground-plan, and arranged for drafts of Greeks, Egyptians, and Jews to be sent to occupy it. The city was intended to be the most magnificent in the world, and, the better to secure this, its erection was left in the hands of Dinocrates, the architect of the temple of Diana at Ephesus. After Alexander’s death in 323 b.c. his empire was divided, and Egypt fell to the lot of Ptolemy, who chose Alexandria as the capital of his kingdom. A short period of confusion followed, but as soon as Ptolemy was settled on the throne, say about 306 b.c., he determined to attract, so far as he was able, learned men of all sorts to his new city; and he at once began the erection of the university buildings on a piece of ground immediately adjoining his palace. The university was ready to be opened somewhere about 300 b.c., and Ptolemy, who wished to secure for its staff the most eminent philosophers of the time, naturally turned to Athens to find them. The great library which was the central feature of the scheme was placed under Demetrius Phalereus, a distinguished Athenian, and so rapidly did it grow that within forty years it (together with the Egyptian annexe) possessed about 600,000 rolls. The mathematical department was placed under Euclid, who was thus the first, as he was one of the most famous, of the mathematicians of the Alexandrian school. It happens that contemporaneously with the foundation of this school the information on which our history is based becomes more ample and certain. Many of the works of the Alexandrian mathematicians are still extant; and we have besides an invaluable treatise by Pappus, described below, in which their best-known treatises are collated, discussed, and criticized. It curiously turns out that just as we begin to be able to speak with confidence on the subject-matter which was taught, we find that our information as to the personality of the teachers becomes vague; and we know very little of the lives of the mathematicians mentioned in this and the next chapter, even the dates at which they lived being frequently in doubt.

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The third century before Christ. Euclid.1 —This century produced three of the greatest mathematicians of antiquity, namely Euclid, Archimedes, and Apollonius. The earliest of these was Euclid. Of his life we know next to nothing, save that he was of Greek descent, and was born about 330 b.c.; he died about 275 b.c. It would appear that he was well acquainted with the Platonic geometry, but he does not seem to have read Aristotle’s works; and these facts are supposed to strengthen the tradition that he was educated at Athens. Whatever may have been his previous training and career, he proved a most successful teacher when settled at Alexandria. He impressed his own individuality on the teaching of the new university to such an extent that to his successors and almost to his contemporaries the name Euclid meant (as it does to us) the book or books he wrote, and not the man himself. Some of the medieval writers went so far as to deny his existence, and with the ingenuity of philologists they explained that the term was only a corruption of ὐκλι a key, and δις geometry. The former word was presumably derived from κλείς. I can only explain the meaning assigned to δις by the conjecture that as the Pythagoreans said that the number two symbolized a line, possibly a schoolman may have thought that it could be taken as indicative of geometry. From the meagre notices of Euclid which have come down to us we find that the saying that there is no royal road in geometry was attributed to Euclid as well as to Menaechmus; but it is an epigrammatic remark which has had many imitators. According to tradition, Euclid was noticeable for his gentleness and modesty. Of his teaching, an anecdote has been preserved. Stobaeus, who is a somewhat doubtful authority, tells us that, when a lad who had just begun geometry asked, “What do I gain by learning all this stuff?” Euclid insisted that knowledge was worth acquiring for its own sake, but made his slave give the boy some coppers, “since,” said he, “he must make a profit out of what he learns.” 1

Besides Loria, book ii, chap. i; Cantor, chaps. xii, xiii; and Gow, pp. 72–86, 195– 221; see the articles Eucleides by A. De Morgan in Smith’s Dictionary of Greek and Roman Biography, London, 1849; the article on Irrational Quantity by A. De Morgan in the Penny Cyclopaedia, London, 1839; Litterargeschichtliche Studien u ¨ber Euklid, by J. L. Heiberg, Leipzig, 1882; and above all Euclid’s Elements, translated with an introduction and commentary by T. L. Heath, 3 volumes, Cambridge, 1908. The latest complete edition of all Euclid’s works is that by J. L. Heiberg and H. Menge, Leipzig, 1883–96.

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Euclid was the author of several works, but his reputation rests mainly on his Elements. This treatise contains a systematic exposition of the leading propositions of elementary metrical geometry (exclusive of conic sections) and of the theory of numbers. It was at once adopted by the Greeks as the standard text-book on the elements of pure mathematics, and it is probable that it was written for that purpose and not as a philosophical attempt to shew that the results of geometry and arithmetic are necessary truths. The modern text1 is founded on an edition or commentary prepared by Theon, the father of Hypatia (circ. 380 a.d.). There is at the Vatican a copy (circ. 1000 a.d.) of an older text, and we have besides quotations from the work and references to it by numerous writers of various dates. From these sources we gather that the definitions, axioms, and postulates were rearranged and slightly altered by subsequent editors, but that the propositions themselves are substantially as Euclid wrote them. As to the matter of the work. The geometrical part is to a large extent a compilation from the works of previous writers. Thus the substance of books i and ii (except perhaps the treatment of parallels) is probably due to Pythagoras; that of book iii to Hippocrates; that of book v to Eudoxus; and the bulk of books iv, vi, xi, and xii to the later Pythagorean or Athenian schools. But this material was rearranged, obvious deductions were omitted (for instance, the proposition that the perpendiculars from the angular points of a triangle on the opposite sides meet in a point was cut out), and in some cases new proofs substituted. Book X, which deals with irrational magnitudes, may be founded on the lost book of Theaetetus; but probably much of it is original, for Proclus says that while Euclid arranged the propositions of Eudoxus he completed many of those of Theaetetus. The whole was presented as a complete and consistent body of theorems. The form in which the propositions are presented, consisting of enunciation, statement, construction, proof, and conclusion, is due to Euclid: so also is the synthetical character of the work, each proof being written out as a logically correct train of reasoning but without any clue to the method by which it was obtained. 1

Most of the modern text-books in English are founded on Simson’s edition, issued in 1758. Robert Simson, who was born in 1687 and died in 1768, was professor of mathematics at the University of Glasgow, and left several valuable works on ancient geometry.

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The defects of Euclid’s Elements as a text-book of geometry have been often stated; the most prominent are these. (i) The definitions and axioms contain many assumptions which are not obvious, and in particular the postulate or axiom about parallel lines is not self-evident.1 (ii) No explanation is given as to the reason why the proofs take the form in which they are presented, that is, the synthetical proof is given but not the analysis by which it was obtained. (iii) There is no attempt made to generalize the results arrived at; for instance, the idea of an angle is never extended so as to cover the case where it is equal to or greater than two right angles: the second half of the thirty-third proposition in the sixth book, as now printed, appears to be an exception, but it is due to Theon and not to Euclid. (iv) The principle of superposition as a method of proof might be used more frequently with advantage. (v) The classification is imperfect. And (vi) the work is unnecessarily long and verbose. Some of those objections do not apply to certain of the recent school editions of the Elements. On the other hand, the propositions in Euclid are arranged so as to form a chain of geometrical reasoning, proceeding from certain almost obvious assumptions by easy steps to results of considerable complexity. The demonstrations are rigorous, often elegant, and not too difficult for a beginner. Lastly, nearly all the elementary metrical (as opposed to the graphical) properties of space are investigated, while the fact that for two thousand years it was the usual text-book on the subject raises a strong presumption that it is not unsuitable for the purpose. On the Continent rather more than a century ago, Euclid was generally superseded by other text-books. In England determined efforts have lately been made with the same purpose, and numerous other works on elementary geometry have been produced in the last decade. The change is too recent to enable us to say definitely what its effect may be. But as far as I can judge, boys who have learnt their geometry on the new system know more facts, but have missed the mental and logical training which was inseparable from a judicious study of Euclid’s treatise. I do not think that all the objections above stated can fairly be urged against Euclid himself. He published a collection of problems, generally known as the Δεδομένα or Data. This contains 95 illustrations of the kind of deductions which frequently have to be made in analysis; 1

We know, from the researches of Lobatschewsky and Riemann, that it is incapable of proof.

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such as that, if one of the data of the problem under consideration be that one angle of some triangle in the figure is constant, then it is legitimate to conclude that the ratio of the area of the rectangle under the sides containing the angle to the area of the triangle is known [prop. 66]. Pappus says that the work was written for those “who wish to acquire the power of solving problems.” It is in fact a gradual series of exercises in geometrical analysis. In short the Elements gave the principal results, and were intended to serve as a training in the science of reasoning, while the Data were intended to develop originality. Euclid also wrote a work called Περὶ Διαιρέσεων or De Divisionibus, known to us only through an Arabic translation which may be itself imperfect.1 This is a collection of 36 problems on the division of areas into parts which bear to one another a given ratio. It is not unlikely that this was only one of several such collections of examples—possibly including the Fallacies and the Porisms—but even by itself it shews that the value of exercises and riders was fully recognized by Euclid. I may here add a suggestion made by De Morgan, whose comments on Euclid’s writings were notably ingenious and informing. From internal evidence he thought it likely that the Elements were written towards the close of Euclid’s life, and that their present form represents the first draft of the proposed work, which, with the exception of the tenth book, Euclid did not live to revise. This opinion is generally discredited, and there is no extrinsic evidence to support it. The geometrical parts of the Elements are so well known that I need do no more than allude to them. Euclid admitted only those constructions which could be made by the use of a ruler and compasses.2 He also excluded practical work and hypothetical constructions. The first four books and book vi deal with plane geometry; the theory of proportion (of any magnitudes) is discussed in book v; and books xi and xii treat of solid geometry. On the hypothesis that the Elements are the first draft of Euclid’s proposed work, it is possible that book xiii 1

R. C. Archibald, Euclid’s Book on Divisions, Cambridge, 1915. The ruler must be of unlimited length and not graduated; the compasses also must be capable of being opened as wide as is desired. Lorenzo Mascheroni (who was born at Castagneta on May 14, 1750, and died at Paris on July 30, 1800) set himself the task to obtain by means of constructions made only with a pair of compasses as many Euclidean results as possible. Mascheroni’s treatise on the geometry of the compass, which was published at Pavia in 1795, is a curious tour de force: he was professor first at Bergamo and afterwards at Pavia, and left numerous minor works. Similar limitations have been proposed by other writers. 2

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is a sort of appendix containing some additional propositions which would have been put ultimately in one or other of the earlier books. Thus, as mentioned above, the first five propositions which deal with a line cut in golden section might be added to the second book. The next seven propositions are concerned with the relations between certain incommensurable lines in plane figures (such as the radius of a circle and the sides of an inscribed regular triangle, pentagon, hexagon, and decagon) which are treated by the methods of the tenth book and as an illustration of them. Constructions of the five regular solids are discussed in the last six propositions, and it seems probable that Euclid and his contemporaries attached great importance to this group of problems. Bretschneider inclined to think that the thirteenth book is a summary of part of the lost work of Aristaeus: but the illustrations of the methods of the tenth book are due most probably to Theaetetus. Books vii, viii, ix, and x of the Elements are given up to the theory of numbers. The mere art of calculation or λογιστική was taught to boys when quite young, it was stigmatized by Plato as childish, and never received much attention from Greek mathematicians; nor was it regarded as forming part of a course of mathematics. We do not know how it was taught, but the abacus certainly played a prominent part in it. The scientific treatment of numbers was called ἀριθμητική, which I have here generally translated as the science of numbers. It had special reference to ratio, proportion, and the theory of numbers. It is with this alone that most of the extant Greek works deal. In discussing Euclid’s arrangement of the subject, we must therefore bear in mind that those who attended his lectures were already familiar with the art of calculation. The system of numeration adopted by the Greeks is described later,1 but it was so clumsy that it rendered the scientific treatment of numbers much more difficult than that of geometry; hence Euclid commenced his mathematical course with plane geometry. At the same time it must be observed that the results of the second book, though geometrical in form, are capable of expression in algebraical language, and the fact that numbers could be represented by lines was probably insisted on at an early stage, and illustrated by concrete examples. This graphical method of using lines to represent numbers possesses the obvious advantage of leading to proofs which are true for all numbers, rational or irrational. It will be noticed that among other propositions in the second book we get geometrical proofs 1

See below, chapter vii.

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of the distributive and commutative laws, of rules for multiplication, and finally geometrical solutions of the equations a(a − x) = x2 , that is x2 +ax−a2 = 0 (Euc. ii, 11), and x2 −ab = 0 (Euc. ii, √14): the solution of the first of these equations is given in the form a2 + ( 12 a)2 − 12 a. The solutions of the equations ax2 − bx + c = 0 and ax2 + bx − c = 0 are given later in Euc. vi, 28 and vi, 29; the cases when a = 1 can be deduced from the identities proved in Euc. ii, 5 and 6, but it is doubtful if Euclid recognized this. The results of the fifth book, in which the theory of proportion is considered, apply to any magnitudes, and therefore are true of numbers as well as of geometrical magnitudes. In the opinion of many writers this is the most satisfactory way of treating the theory of proportion on a scientific basis; and it was used by Euclid as the foundation on which he built the theory of numbers. The theory of proportion given in this book is believed to be due to Eudoxus. The treatment of the same subject in the seventh book is less elegant, and is supposed to be a reproduction of the Pythagorean teaching. This double discussion of proportion is, as far as it goes, in favour of the conjecture that Euclid did not live to revise the work. In books vii, viii, and ix Euclid discusses the theory of rational numbers. He commences the seventh book with some definitions founded on the Pythagorean notation. In propositions 1 to 3 he shews that if, in the usual process for finding the greatest common measure of two numbers, the last divisor be unity, the numbers must be prime; and he thence deduces the rule for finding their G.C.M. Propositions 4 to 22 include the theory of fractions, which he bases on the theory of proportion; among other results he shews that ab = ba [prop. 16]. In propositions 23 to 34 he treats of prime numbers, giving many of the theorems in modern text-books on algebra. In propositions 35 to 41 he discusses the least common multiple of numbers, and some miscellaneous problems. The eighth book is chiefly devoted to numbers in continued proportion, that is, in a geometrical progression; and the cases where one or more is a product, square, or cube are specially considered. In the ninth book Euclid continues the discussion of geometrical progressions, and in proposition 35 he enunciates the rule for the summation of a series of n terms, though the proof is given only for the case where n is equal to 4. He also develops the theory of primes, shews that the number of primes is infinite [prop. 20], and discusses the properties

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of odd and even numbers. He concludes by shewing that a number of the form 2n−1 (2n − 1), where 2n − 1 is a prime, is a “perfect” number [prop. 36]. In the tenth book Euclid deals with certain irrational magnitudes; and, since the Greeks possessed no symbolism for surds, he was forced to adopt a geometrical representation. Propositions 1 to 21 deal generally with incommensurable magnitudes. The rest of the book, namely, propositions 22 to 117, is devoted to the discussion √√ √of every possible variety of lines which can be represented by ( a ± b), where a and b denote commensurable lines. There are twenty-five species of such lines, and that Euclid could detect and classify them all is in the opinion of so competent an authority as Nesselmann the most striking illustration of his genius. No further advance in the theory of incommensurable magnitudes was made until the subject was taken up by Leonardo and Cardan after an interval of more than a thousand years. In the last proposition of the tenth book [prop. 117] the side and diagonal of a square are proved to be incommensurable. The proof is so short and easy that I may quote it. If possible let the side be to the diagonal in a commensurable ratio, namely, that of two integers, a and b. Suppose this ratio reduced to its lowest terms so that a and b have no common divisor other than unity, that is, they are prime to one another. Then (by Euc. i, 47) b2 = 2a2 ; therefore b2 is an even number; therefore b is an even number; hence, since a is prime to b, a must be an odd number. Again, since it has been shewn that b is an even number, b may be represented by 2n; therefore (2n)2 = 2a2 ; therefore a2 = 2n2 ; therefore a2 is an even number; therefore a is an even number. Thus the same number a must be both odd and even, which is absurd; therefore the side and diagonal are incommensurable. Hankel believes that this proof was due to Pythagoras, and this is not unlikely. This proposition is also proved in another way in Euc. x, 9, and for this and other reasons it is now usually believed to be an interpolation by some commentator on the Elements. In addition to the Elements and the two collections of riders above mentioned (which are extant) Euclid wrote the following books on geometry: (i) an elementary treatise on conic sections in four books; (ii) a book on surface loci, probably confined to curves on the cone and cylinder; (iii) a collection of geometrical fallacies, which were to be used as exercises in the detection of errors; and (iv) a treatise on porisms arranged in three books. All of these are lost, but the work on porisms was discussed at such length by Pappus, that some writers have thought

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it possible to restore it. In particular, Chasles in 1860 published what he considered to be substantially a reproduction of it. In this will be found the conceptions of cross ratios and projection, and those ideas of modern geometry which were used so extensively by Chasles and other writers of the nineteenth century. It should be realized, however, that the statements of the classical writers concerning this book are either very brief or have come to us only in a mutilated form, and De Morgan frankly says that he found them unintelligible, an opinion in which most of those who read them will, I think, concur. Euclid published a book on optics, treated geometrically, which contains 61 propositions founded on 12 assumptions. It commences with the assumption that objects are seen by rays emitted from the eye in straight lines, “for if light proceeded from the object we should not, as we often do, fail to perceive a needle on the floor.” A work called Catoptrica is also attributed to him by some of the older writers; the text is corrupt and the authorship doubtful; it consists of 31 propositions dealing with reflexions in plane, convex, and concave mirrors. The geometry of both books is Euclidean in form. Euclid has been credited with an ingenious demonstration1 of the principle of the lever, but its authenticity is doubtful. He also wrote the Phaenomena, a treatise on geometrical astronomy. It contains references to the work of Autolycus2 and to some book on spherical geometry by an unknown writer. Pappus asserts that Euclid also composed a book on the elements of music: this may refer to the Sectio Canonis, which is by Euclid, and deals with musical intervals. To these works I may add the following little problem, which occurs in the Palatine Anthology and is attributed by tradition to Euclid. “A mule and a donkey were going to market laden with wheat. The mule said, ‘If you gave me one measure I should carry twice as much as you, but if I gave you one we should bear equal burdens.’ Tell me, learned geometrician, what were their burdens.” It is impossible to say whether the question is due to Euclid, but there is nothing improbable in the suggestion. It will be noticed that Euclid dealt only with magnitudes, and did 1 It is given (from the Arabic) by F. Woepcke in the Journal Asiatique, series 4, vol. xviii, October 1851, pp. 225–232. 2 Autolycus lived at Pitane in Aeolis and flourished about 330 b.c. His two works on astronomy, containing 43 propositions, are said to be the oldest extant Greek mathematical treatises. They exist in manuscript at Oxford. They were edited, with a Latin translation, by F. Hultsch, Leipzig, 1885.

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not concern himself with their numerical measures, but it would seem from the works of Aristarchus and Archimedes that this was not the case with all the Greek mathematicians of that time. As one of the works of the former is extant it will serve as another illustration of Greek mathematics of this period. Aristarchus. Aristarchus of Samos, born in 310 b.c. and died in 250 b.c., was an astronomer rather than a mathematician. He asserted, at any rate as a working hypothesis, that the sun was the centre of the universe, and that the earth revolved round the sun. This view, in spite of the simple explanation it afforded of various phenomena, was generally rejected by his contemporaries. But his propositions1 on the measurement of the sizes and distances of the sun and moon were accurate in principle, and his results were accepted by Archimedes in his Ψαμμίτης, mentioned below, as approximately correct. There are 19 theorems, of which I select the seventh as a typical illustration, because it shews the way in which the Greeks evaded the difficulty of finding the numerical value of surds. Aristarchus observed the angular distance between the moon when dichotomized and the sun, and found it to be twenty-nine thirtieths of a right angle. It is actually about 89◦ 210 , but of course his instruments were of the roughest description. He then proceeded to shew that the distance of the sun is greater than eighteen and less than twenty times the distance of the moon in the following manner. Let S be the sun, E the earth, and M the moon. Then when the moon is dichotomized, that is, when the bright part which we see is exactly a half-circle, the angle between M S and M E is a right angle. With E as centre, and radii ES and EM describe circles, as in the figure below. Draw EA perpendicular to ES. Draw EF bisecting the angle AES, and EG bisecting the angle AEF , as in the figure. Let 1 th EM (produced) cut AF in H. The angle AEM is by hypothesis 30 of a right angle. Hence we have 1 angle AEG : angle AEH = 14 rt. ∠ : 30 rt. ∠ = 15 : 2, ∴ AG : AH [ = tan AEG : tan AEH] > 15 : 2.

1

(α)

Περὶ μεγέθων καὶ ἀποστημάτων `Ηλίου καὶ Σελήνης, edited by E. Nizze, Stralsund, 1856. Latin translations were issued by F. Commandino in 1572 and by J. Wallis in 1688; and a French translation was published by F. d’Urban in 1810 and 1823.

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G

H

A

M

S

E

Again F G2 : AG2 = EF 2 : EA2 (Euc. vi, 3) = 2 : 1 (Euc. i, 47), ∴ F G2 : AG2 > 49 : 25, ∴ F G : AG > 7 : 5, ∴ AF : AG > 12 : 5, ∴ AE : AG > 12 : 5.

(β)

Compounding the ratios (α) and (β), we have AE : AH > 18 : 1. But the triangles EM S and EAH are similar, ∴ ES : EM > 18 : 1. I will leave the second half of the proposition to amuse any reader who may care to prove it: the analysis is straightforward. In a somewhat similar way Aristarchus found the ratio of the radii of the sun, earth, and moon. We know very little of Conon and Dositheus, the immediate successors of Euclid at Alexandria, or of their contemporaries Zeuxippus

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and Nicoteles, who most likely also lectured there, except that Archimedes, who was a student at Alexandria probably shortly after Euclid’s death, had a high opinion of their ability and corresponded with the three first mentioned. Their work and reputation has been completely overshadowed by that of Archimedes. Archimedes.1 Archimedes, who probably was related to the royal family at Syracuse, was born there in 287 b.c. and died in 212 b.c. He went to the university of Alexandria and attended the lectures of Conon, but, as soon as he had finished his studies, returned to Sicily where he passed the remainder of his life. He took no part in public affairs, but his mechanical ingenuity was astonishing, and, on any difficulties which could be overcome by material means arising, his advice was generally asked by the government. Archimedes, like Plato, held that it was undesirable for a philosopher to seek to apply the results of science to any practical use; but in fact he did introduce a large number of new inventions. The stories of the detection of the fraudulent goldsmith and of the use of burningglasses to destroy the ships of the Roman blockading squadron will recur to most readers. Perhaps it is not as well known that Hiero, who had built a ship so large that he could not launch it off the slips, applied to Archimedes. The difficulty was overcome by means of an apparatus of cogwheels worked by an endless screw, but we are not told exactly how the machine was used. It is said that it was on this occasion, in acknowledging the compliments of Hiero, that Archimedes made the well-known remark that had he but a fixed fulcrum he could move the earth. Most mathematicians are aware that the Archimedean screw was another of his inventions. It consists of a tube, open at both ends, and bent into the form of a spiral like a corkscrew. If one end be immersed in water, and the axis of the instrument (i.e. the axis of the cylinder on the surface of which the tube lies) be inclined to the vertical at a sufficiently big angle, and the instrument turned round it, the water 1

Besides Loria, book ii, chap. iii, Cantor, chaps. xiv, xv, and Gow, pp. 221– 244, see Quaestiones Archimedeae, by J. L. Heiberg, Copenhagen, 1879; and Marie, vol. i, pp. 81–134. The best editions of the extant works of Archimedes are those by J. L. Heiberg, in 3 vols., Leipzig, 1880–81, and by Sir Thomas L. Heath, Cambridge, 1897. In 1906 a manuscript, previously unknown, was discovered at Constantinople, containing propositions on hydrostatics and on methods; see Eine neue Schrift des Archimedes, by J. L. Heiberg and H. G. Zeuthen, Leipzig, 1907, and the Method of Archimedes, by Sir Thomas L. Heath, Cambridge, 1912.

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will flow along the tube and out at the other end. In order that it may work, the inclination of the axis of the instrument to the vertical must be greater than the pitch of the screw. It was used in Egypt to drain the fields after an inundation of the Nile, and was also frequently applied to take water out of the hold of a ship. The story that Archimedes set fire to the Roman ships by means of burning-glasses and concave mirrors is not mentioned till some centuries after his death, and is generally rejected. The mirror of Archimedes is said to have been made in the form of a hexagon surrounded by rings of polygons; and Buffon1 in 1747 contrived, by the use of a single composite mirror made on this model, to set fire to wood at a distance of 150 feet, and to melt lead at a distance of 140 feet. This was in April and as far north as Paris, so in a Sicilian summer the use of several such mirrors might be a serious annoyance to a blockading fleet, if the ships were sufficiently near. It is perhaps worth mentioning that a similar device is said to have been used in the defence of Constantinople in 514 a.d., and is alluded to by writers who either were present at the siege or obtained their information from those who were engaged in it. But whatever be the truth as to this story, there is no doubt that Archimedes devised the catapults which kept the Romans, who were then besieging Syracuse, at bay for a considerable time. These were constructed so that the range could be made either short or long at pleasure, and so that they could be discharged through a small loophole without exposing the artillery-men to the fire of the enemy. So effective did they prove that the siege was turned into a blockade, and three years elapsed before the town was taken. Archimedes was killed during the sack of the city which followed its capture, in spite of the orders, given by the consul Marcellus who was in command of the Romans, that his house and life should be spared. It is said that a soldier entered his study while he was regarding a geometrical diagram drawn in sand on the floor, which was the usual way of drawing figures in classical times. Archimedes told him to get off the diagram, and not spoil it. The soldier, feeling insulted at having orders given to him and ignorant of who the old man was, killed him. According to another and more probable account, the cupidity of the troops was excited by seeing his instruments, constructed of polished brass which they supposed to be made of gold. 1

101.

See M´emoires de l’acad´emie royale des sciences for 1747, Paris, 1752, pp. 82–

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The Romans erected a splendid tomb to Archimedes, on which was engraved (in accordance with a wish he had expressed) the figure of a sphere inscribed in a cylinder, in commemoration of the proof he had given that the volume of a sphere was equal to two-thirds that of the circumscribing right cylinder, and its surface to four times the area of a great circle. Cicero1 gives a charming account of his efforts (which were successful) to rediscover the tomb in 75 b.c. It is difficult to explain in a concise form the works or discoveries of Archimedes, partly because he wrote on nearly all the mathematical subjects then known, and partly because his writings are contained in a series of disconnected monographs. Thus, while Euclid aimed at producing systematic treatises which could be understood by all students who had attained a certain level of education, Archimedes wrote a number of brilliant essays addressed chiefly to the most educated mathematicians of the day. The work for which he is perhaps now best known is his treatment of the mechanics of solids and fluids; but he and his contemporaries esteemed his geometrical discoveries of the quadrature of a parabolic area and of a spherical surface, and his rule for finding the volume of a sphere as more remarkable; while at a somewhat later time his numerous mechanical inventions excited most attention. (i) On plane geometry the extant works of Archimedes are three in number, namely, (a) the Measure of the Circle, (b) the Quadrature of the Parabola, and (c) one on Spirals. (a) The Measure of the Circle contains three propositions. In the first proposition Archimedes proves that the area is the same as that of a right-angled triangle whose sides are equal respectively to the radius a and the circumference of the circle, i.e. the area is equal to 12 a(2πa). In the second proposition he shows that πa2 : (2a)2 = 11 : 14 very nearly; and next, in the third proposition, that π is less than 3 71 and greater than 3 10 . These theorems are of course proved geometrically. 71 To demonstrate the two latter propositions, he inscribes in and circumscribes about a circle regular polygons of ninety-six sides, calculates their perimeters, and then assumes the circumference of the circle to lie between them: this leads to the result 6336/2017 14 < π < 14688/4673 12 , from which he deduces the limits given above. It would seem from the proof that he had some (at present unknown) method of extracting the square roots of numbers approximately. The table which he formed of the numerical values of the chords of a circle is essentially a table of 1

See his Tusculanarum Disputationum, v. 23.

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natural sines, and may have suggested the subsequent work on these lines of Hipparchus and Ptolemy. (b) The Quadrature of the Parabola contains twenty-four propositions. Archimedes begins this work, which was sent to Dositheus, by establishing some properties of conics [props. 1–5]. He then states correctly the area cut off from a parabola by any chord, and gives a proof which rests on a preliminary mechanical experiment of the ratio of areas which balance when suspended from the arms of a lever [props. 6–17]; and, lastly, he gives a geometrical demonstration of this result [props. 18–24]. The latter is, of course, based on the method of exhaustions, but for brevity I will, in quoting it, use the method of limits. P

V

M

Q

Let the area of the parabola (see figure above) be bounded by the chord P Q. Draw V M the diameter to the chord P Q, then (by a previous proposition), V is more remote from P Q than any other point in the arc P V Q. Let the area of the triangle P V Q be denoted by 4. In the segments bounded by V P and V Q inscribe triangles in the same way as the triangle P V Q was inscribed in the given segment. Each of these triangles is (by a previous proposition of his) equal to 81 4, and their sum is therefore 41 4. Similarly in the four segments left inscribe

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1 triangles; their sum will be 16 4. Proceeding in this way the area of the given segment is shown to be equal to the limit of

4+

4 4 4 + + ··· + n + ··· , 4 16 4

when n is indefinitely large. The problem is therefore reduced to finding the sum of a geometrical series. This he effects as follows. Let A, B, C, . . . , J, K be a series of magnitudes such that each is one-fourth of that which precedes it. Take magnitudes b, c, . . . , k equal respectively to 13 B, 13 C, . . . , 13 K. Then B + b = 13 A,

C + c = 31 B,

...,

K + k = 13 J.

Hence (B + C + . . . + K) + (b + c + . . . + k) = 13 (A + B + . . . + J); but, by hypothesis, (b + c + . . . + j + k) = 31 (B + C + . . . + J) + 13 K; ∴ (B + C + . . . + K) + 13 K = 13 A. ∴ A + B + C + . . . + K = 34 A − 13 K. Hence the sum of these magnitudes exceeds four times the third of the largest of them by one-third of the smallest of them. Returning now to the problem of the quadrature of the parabola A stands for ∆, and ultimately K is indefinitely small; therefore the area of the parabolic segment is four-thirds that of the triangle P V Q, or two-thirds that of a rectangle whose base is P Q and altitude the distance of V from P Q. While discussing the question of quadratures it may be added that in the fifth and sixth propositions of his work on conoids and spheroids he determined the area of an ellipse. (c) The work on Spirals contains twenty-eight propositions on the properties of the curve now known as the spiral of Archimedes. It was sent to Dositheus at Alexandria accompanied by a letter, from which it appears that Archimedes had previously sent a note of his results to Conon, who had died before he had been able to prove them. The spiral is defined by saying that the vectorial angle and radius vector both increase uniformly, hence its equation is r = cθ. Archimedes finds most of its properties, and determines the area inclosed between the curve and two radii vectores. This he does (in effect) by saying, in the language of the infinitesimal calculus, that an element of area is > 21 r2 dθ and < 12 (r + dr)2 dθ: to effect the sum of the elementary

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areas he gives two lemmas in which he sums (geometrically) the series a2 + (2a)2 + (3a)2 + . . . + (na)2 [prop. 10], and a + 2a + 3a + . . . + na [prop. 11]. (d ) In addition to these he wrote a small treatise on geometrical methods, and works on parallel lines, triangles, the properties of rightangled triangles, data, the heptagon inscribed in a circle, and systems of circles touching one another ; possibly he wrote others too. These are all lost, but it is probable that fragments of four of the propositions in the last-mentioned work are preserved in a Latin translation from an Arabic manuscript entitled Lemmas of Archimedes. (ii) On geometry of three dimensions the extant works of Archimedes are two in number, namely (a), the Sphere and Cylinder, and (b) Conoids and Spheroids. (a) The Sphere and Cylinder contains sixty propositions arranged in two books. Archimedes sent this like so many of his works to Dositheus at Alexandria; but he seems to have played a practical joke on his friends there, for he purposely misstated some of his results “to deceive those vain geometricians who say they have found everything, but never give their proofs, and sometimes claim that they have discovered what is impossible.” He regarded this work as his masterpiece. It is too long for me to give an analysis of its contents, but I remark in passing that in it he finds expressions for the surface and volume of a pyramid, of a cone, and of a sphere, as well as of the figures produced by the revolution of polygons inscribed in a circle about a diameter of the circle. There are several other propositions on areas and volumes of which perhaps the most striking is the tenth proposition of the second book, namely, that “of all spherical segments whose surfaces are equal the hemisphere has the greatest volume.” In the second proposition of the second book he enunciates the remarkable theorem that a line of length a can be divided so that a − x : b = 4a2 : 9x2 (where b is a given length), only if b be less than 31 a; that is to say, the cubic equation x3 − ax2 + 49 a2 b = 0 can have a real and positive root only if a be greater than 3b. This proposition was required to complete his solution of the problem to divide a given sphere by a plane so that the volumes of the segments should be in a given ratio. One very simple cubic equation occurs in the Arithmetic of Diophantus, but with that exception no such equation appears again in the history of European mathematics for more than a thousand years. (b) The Conoids and Spheroids contains forty propositions on quadrics of revolution (sent to Dositheus in Alexandria) mostly concerned

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with an investigation of their volumes. (c) Archimedes also wrote a treatise on certain semi-regular polyhedrons, that is, solids contained by regular but dissimilar polygons. This is lost, but references to it are given by Pappus. (iii) On arithmetic Archimedes wrote two papers. One (addressed to Zeuxippus) was on the principles of numeration; this is now lost. The other (addressed to Gelon) was called Ψαμμίτης (the sand-reckoner ), and in this he meets an objection which had been urged against his first paper. The object of the first paper had been to suggest a convenient system by which numbers of any magnitude could be represented; and it would seem that some philosophers at Syracuse had doubted whether the system was practicable. Archimedes says people talk of the sand on the Sicilian shore as something beyond the power of calculation, but he can estimate it; and, further, he will illustrate the power of his method by finding a superior limit to the number of grains of sand which would fill the whole universe, i.e. a sphere whose centre is the earth, and radius the distance of the sun. He begins by saying that in ordinary Greek nomenclature it was only possible to express numbers from 1 up to 108 : these are expressed in what he says he may call units of the first order. If 108 be termed a unit of the second order, any number from 108 to 1016 can be expressed as so many units of the second order plus so many units of the first order. If 1016 be a unit of the third order any number up to 1024 can be then expressed, and so on. Assuming that 1 th 10,000 grains of sand occupy a sphere whose radius is not less than 80 of a finger-breadth, and that the diameter of the universe is not greater than 1010 stadia, he finds that the number of grains of sand required to fill the solar universe is less than 1051 . Probably this system of numeration was suggested merely as a scientific curiosity. The Greek system of numeration with which we are acquainted had been only recently introduced, most likely at Alexandria, and was sufficient for all the purposes for which the Greeks then required numbers; and Archimedes used that system in all his papers. On the other hand, it has been conjectured that Archimedes and Apollonius had some symbolism based on the decimal system for their own investigations, and it is possible that it was the one here sketched out. The units suggested by Archimedes form a geometrical progression, having 108 for the radix. He incidentally adds that it will be convenient to remember that the product of the mth and nth terms of a geometrical progression, whose first term is unity, is equal to the (m + n)th

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term of the series, that is, that rm × rn = rm+n . To these two arithmetical papers I may add the following celebrated problem1 which he sent to the Alexandrian mathematicians. The sun had a herd of bulls and cows, all of which were either white, grey, dun, or piebald: the number of piebald bulls was less than the number of white bulls by 5/6ths of the number of grey bulls, it was less than the number of grey bulls by 9/20ths of the number of dun bulls, and it was less than the number of dun bulls by 13/42nds of the number of white bulls; the number of white cows was 7/12ths of the number of grey cattle (bulls and cows), the number of grey cows was 9/20ths of the number of dun cattle, the number of dun cows was 11/30ths of the number of piebald cattle, and the number of piebald cows was 13/42nds of the number of white cattle. The problem was to find the composition of the herd. The problem is indeterminate, but the solution in lowest integers is white bulls, . . . . . grey bulls, . . . . . . . dun bulls, . . . . . . . piebald bulls, . . . .

10,366,482; 7,460,514; 7,358,060; 4,149,387;

white cows,. . . . . . grey cows, . . . . . . . dun cows, . . . . . . . piebald cows, . . . .

7,206,360; 4,893,246; 3,515,820; 5,439,213.

In the classical solution, attributed to Archimedes, these numbers are multiplied by 80. Nesselmann believes, from internal evidence, that the problem has been falsely attributed to Archimedes. It certainly is unlike his extant work, but it was attributed to him among the ancients, and is generally thought to be genuine, though possibly it has come down to us in a modified form. It is in verse, and a later copyist has added the additional conditions that the sum of the white and grey bulls shall be a square number, and the sum of the piebald and dun bulls a triangular number. It is perhaps worthy of note that in the enunciation the fractions are represented as a sum of fractions whose numerators are unity: thus Archimedes wrote 1/7+1/6 instead of 13/42, in the same way as Ahmes would have done. (iv) On mechanics the extant works of Archimedes are two in number, namely, (a) his Mechanics, and (c) his Hydrostatics. 1

See a memoir by B. Krumbiegel and A. Amthor, Zeitschrift f¨ ur Mathematik, Abhandlungen zur Geschichte der Mathematik, Leipzig, vol. xxv, 1880, pp. 121–136, 153–171.

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(a) The Mechanics is a work on statics with special reference to the equilibrium of plane laminas and to properties of their centres of gravity; it consists of twenty-five propositions in two books. In the first part of book i, most of the elementary properties of the centre of gravity are proved [props. 1–8]; and in the remainder of book i, [props. 9– 15] and in book ii the centres of gravity of a variety of plane areas, such as parallelograms, triangles, trapeziums, and parabolic areas are determined. As an illustration of the influence of Archimedes on the history of mathematics, I may mention that the science of statics rested on his theory of the lever until 1586, when Stevinus published his treatise on statics. His reasoning is sufficiently illustrated by an outline of his proof for the case of two weights, P and Q, placed at their centres of gravity, A and B, on a weightless bar AB. He wants to shew that the centre of gravity of P and Q is at a point O on the bar such that P.OA = Q.OB. L

H A

K O

B

On the line AB (produced if necessary) take points H and K, so that HB = BK = AO; and a point L so that LA = OB. It follows that LH will be bisected at A, HK at B, and LK at O; also LH : HK = AH : HB = OB : AO = P : Q. Hence, by a previous proposition, we may consider that the effect of P is the same as that of a heavy uniform bar LH of weight P , and the effect of Q is the same as that of a similar heavy uniform bar HK of weight Q. Hence the effect of the weights is the same as that of a heavy uniform bar LK. But the centre of gravity of such a bar is at its middle point O. (b) Archimedes also wrote a treatise on levers and perhaps, on all the mechanical machines. The book is lost, but we know from Pappus that it contained a discussion of how a given weight could be moved with a given power. It was in this work probably that Archimedes discussed the theory of a certain compound pulley consisting of three or more simple pulleys which he had invented, and which was used in some public works in Syracuse. It is well known1 that he boasted that, if he had but a fixed fulcrum, he could move the whole earth; and a 1

See above, p. 53.

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commentator of later date says that he added he would do it by using a compound pulley. (c) His work on floating bodies contains nineteen propositions in two books, and was the first attempt to apply mathematical reasoning to hydrostatics. The story of the manner in which his attention was directed to the subject is told by Vitruvius. Hiero, the king of Syracuse, had given some gold to a goldsmith to make into a crown. The crown was delivered, made up, and of the proper weight, but it was suspected that the workman had appropriated some of the gold, replacing it by an equal weight of silver. Archimedes was thereupon consulted. Shortly afterwards, when in the public baths, he noticed that his body was pressed upwards by a force which increased the more completely he was immersed in the water. Recognising the value of the observation, he rushed out, just as he was, and ran home through the streets, shouting εὕρηκα, εὕρηκα, “I have found it, I have found it.” There (to follow a later account) on making accurate experiments he found that when equal weights of gold and silver were weighed in water they no longer appeared equal: each seemed lighter than before by the weight of the water it displaced, and as the silver was more bulky than the gold its weight was more diminished. Hence, if on a balance he weighed the crown against an equal weight of gold and then immersed the whole in water, the gold would outweigh the crown if any silver had been used in its construction. Tradition says that the goldsmith was found to be fraudulent. Archimedes began the work by proving that the surface of a fluid at rest is spherical, the centre of the sphere being at the centre of the earth. He then proved that the pressure of the fluid on a body, wholly or partially immersed, is equal to the weight of the fluid displaced; and thence found the position of equilibrium of a floating body, which he illustrated by spherical segments and paraboloids of revolution floating on a fluid. Some of the latter problems involve geometrical reasoning of considerable complexity. The following is a fair specimen of the questions considered. A solid in the shape of a paraboloid of revolution of height h and latus rectum 4a floats in water, with its vertex immersed and its base wholly above the surface. If equilibrium be possible when the axis is not vertical, then the density of the body must be less than (h − 3a)2 /h3 [book ii, prop. 4]. When it is recollected that Archimedes was unacquainted with trigonometry or analytical geometry, the fact that he could discover and prove a proposition such as that just quoted will serve as an illustration

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of his powers of analysis. It will be noticed that the mechanical investigations of Archimedes were concerned with statics. It may be added that though the Greeks attacked a few problems in dynamics, they did it with but indifferent success: some of their remarks were acute, but they did not sufficiently realise that the fundamental facts on which the theory must be based can be established only by carefully devised observations and experiments. It was not until the time of Galileo and Newton that this was done. (v) We know, both from occasional references in his works and from remarks by other writers, that Archimedes was largely occupied in astronomical observations. He wrote a book, Περὶ Σφειροποιίας, on the construction of a celestial sphere, which is lost; and he constructed a sphere of the stars, and an orrery. These, after the capture of Syracuse, were taken by Marcellus to Rome, and were preserved as curiosities for at least two or three hundred years. This mere catalogue of his works will show how wonderful were his achieveC ments; but no one who has not actually read some of D his writings can form a just appreciation of his extraorA B dinary ability. This will be still further increased if we recollect that the only principles used by Archimedes, in addition to those contained in Euclid’s Elements and Conic sections, are that of all lines like ACB, ADB, . . . connecting two points A and B, the straight line is the shortest, and of the curved lines, the inner one ADB is shorter than the outer one ACB; together with two similar statements for space of three dimensions. In the old and medieval world Archimedes was reckoned as the first of mathematicians, but possibly the best tribute to his fame is the fact that those writers who have spoken most highly of his work and ability are those who have been themselves the most distinguished men of their own generation. Apollonius.1 The third great mathematician of this century 1

In addition to Zeuthen’s work and the other authorities mentioned in the footnote on p. 41, see Litterargeschichtliche Studien u ¨ber Euklid, by J. L. Heiberg, Leipzig, 1882. Editions of the extant works of Apollonius were issued by

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was Apollonius of Perga, who is chiefly celebrated for having produced a systematic treatise on the conic sections which not only included all that was previously known about them, but immensely extended the knowledge of these curves. This work was accepted at once as the standard text-book on the subject, and completely superseded the previous treatises of Menaechmus, Aristaeus, and Euclid which until that time had been in general use. We know very little of Apollonius himself. He was born about 260 b.c., and died about 200 b.c. He studied in Alexandria for many years, and probably lectured there; he is represented by Pappus as “vain, jealous of the reputation of others, and ready to seize every opportunity to depreciate them.” It is curious that while we know next to nothing of his life, or of that of his contemporary Eratosthenes, yet their nicknames, which were respectively epsilon and beta, have come down to us. Dr. Gow has ingeniously suggested that the lecture rooms at Alexandria were numbered, and that they always used the rooms numbered 5 and 2 respectively. Apollonius spent some years at Pergamum in Pamphylia, where a university had been recently established and endowed in imitation of that at Alexandria. There he met Eudemus and Attalus, to whom he subsequently sent each book of his conics as it came out with an explanatory note. He returned to Alexandria, and lived there till his death, which was nearly contemporaneous with that of Archimedes. In his great work on conic sections Apollonius so thoroughly investigated the properties of these curves that he left but little for his successors to add. But his proofs are long and involved, and I think most readers will be content to accept a short analysis of his work, and the assurance that his demonstrations are valid. Dr. Zeuthen believes that many of the properties enunciated were obtained in the first instance by the use of co-ordinate geometry, and that the demonstrations were translated subsequently into geometrical form. If this be so, we must suppose that the classical writers were familiar with some branches of analytical geometry—Dr. Zeuthen says the use of orthogonal and oblique co-ordinates, and of transformations depending on abridged notation—that this knowledge was confined to a limited school, and was finally lost. This is a mere conjecture and is unsupported by any direct evidence, but it has been accepted by some writers J. L. Heiberg in two volumes, Leipzig, 1890, 1893; and by E. Halley, Oxford, 1706 and 1710: an edition of the conics was published by T. L. Heath, Cambridge, 1896.

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as affording an explanation of the extent and arrangement of the work. The treatise contained about four hundred propositions, and was divided into eight books; we have the Greek text of the first four of these, and we also possess copies of the commentaries by Pappus and Eutocius on the whole work. In the ninth century an Arabic translation was made of the first seven books, which were the only ones then extant; we have two manuscripts of this version. The eighth book is lost. In the letter to Eudemus which accompanied the first book Apollonius says that he undertook the work at the request of Naucrates, a geometrician who had been staying with him at Alexandria, and, though he had given some of his friends a rough draft of it, he had preferred to revise it carefully before sending it to Pergamum. In the note which accompanied the next book, he asks Eudemus to read it and communicate it to others who can understand it, and in particular to Philonides, a certain geometrician whom the author had met at Ephesus. The first four books deal with the elements of the subject, and of these the first three are founded on Euclid’s previous work (which was itself based on the earlier treatises by Menaechmus and Aristaeus). Heracleides asserts that much of the matter in these books was stolen from an unpublished work of Archimedes, but a critical examination by Heiberg has shown that this is improbable.

P

A

M

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Apollonius begins by defining a cone on a circular base. He then investigates the different plane sections of it, and shows that they are divisible into three kinds of curves which he calls ellipses, parabolas, and hyperbolas. He proves the proposition that, if A, A0 be the vertices of a conic, and if P be any point on it, and P M the perpendicular drawn from P on AA0 , then (in the usual notation) the ratio M P 2 : AM . M A0 is constant in an ellipse or hyperbola, and the ratio M P 2 : AM is constant in a parabola. These are the characteristic properties on which almost all the rest of the work is based. He next shows that, if A be the vertex, l the latus rectum, and if AM and M P be the abscissa and ordinate of any point on a conic (see above figure), then M P 2 is less than, equal to, or greater than l . AM according as the conic is an ellipse, parabola, or hyperbola; hence the names which he gave to the curves and by which they are still known. He had no idea of the directrix, and was not aware that the parabola had a focus, but, with the exception of the propositions which involve these, his first three books contain most of the propositions which are found in modern text-books. In the fourth book he develops the theory of lines cut harmonically, and treats of the points of intersection of systems of conics. In the fifth book he commences with the theory of maxima and minima; applies it to find the centre of curvature at any point of a conic, and the evolute of the curve; and discusses the number of normals which can be drawn from a point to a conic. In the sixth book he treats of similar conics. The seventh and eighth books were given up to a discussion of conjugate diameters; the latter of these was conjecturally restored by E. Halley in his edition of 1710. The verbose explanations make the book repulsive to most modern readers; but the arrangement and reasoning are unexceptional, and it has been not unfitly described as the crown of Greek geometry. It is the work on which the reputation of Apollonius rests, and it earned for him the name of “the great geometrician.” Besides this immense treatise he wrote numerous shorter works; of course the books were written in Greek, but they are usually referred to by their Latin titles: those about which we now know anything are enumerated below. He was the author of a work on the problem “given two co-planar straight lines Aa and Bb, drawn through fixed points A and B; to draw a line Oab from a given point O outside them cutting them in a and b, so that Aa shall be to Bb in a given ratio.” He reduced the question to seventy-seven separate cases and gave an appropriate solution, with the aid of conics, for each case; this was published by

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E. Halley (translated from an Arabic copy) in 1706. He also wrote a treatise De Sectione Spatii (restored by E. Halley in 1706) on the same problem under the condition that the rectangle Aa . Bb was given. He wrote another entitled De Sectione Determinata (restored by R. Simson in 1749), dealing with problems such as to find a point P in a given straight line AB, so that P A2 shall be to P B in a given ratio. He wrote another De Tactionibus (restored by Vieta in 1600) on the construction of a circle which shall touch three given circles. Another work was his De Inclinationibus (restored by M. Ghetaldi in 1607) on the problem to draw a line so that the intercept between two given lines, or the circumferences of two given circles, shall be of a given length. He was also the author of a treatise in three books on plane loci, De Locis Planis (restored by Fermat in 1637, and by R. Simson in 1746), and of another on the regular solids. And, lastly, he wrote a treatise on unclassed incommensurables, being a commentary on the tenth book of Euclid. It is believed that in one or more of the lost books he used the method of conical projections. Besides these geometrical works he wrote on the methods of arithmetical calculation. All that we know of this is derived from some remarks of Pappus. Friedlein thinks that it was merely a sort of readyreckoner. It seems, however, more probable that Apollonius here suggested a system of numeration similar to that proposed by Archimedes, but proceeding by tetrads instead of octads, and described a notation for it. It will be noticed that our modern notation goes by hexads, a million = 106 , a billion = 1012 , a trillion = 1018 , etc. It is not impossible that Apollonius also pointed out that a decimal system of notation, involving only nine symbols, would facilitate numerical multiplications. Apollonius was interested in astronomy, and wrote a book on the stations and regressions of the planets of which Ptolemy made some use in writing the Almagest. He also wrote a treatise on the use and theory of the screw in statics. This is a long list, but I should suppose that most of these works were short tracts on special points. Like so many of his predecessors, he too gave a construction for finding two mean proportionals between two given lines, and thereby duplicating the cube. It was as follows. Let OA and OB be the given lines. Construct a rectangle OADB, of which they are adjacent sides. Bisect AB in C. Then, if with C as centre we can describe a circle cutting OA produced in a, and cutting OB produced in b, so that aDb shall be a straight line, the problem is effected. For it is easily shewn

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that Oa . Aa + CA2 = Ca2 . Ob . Bb + CB 2 = Cb2 . Oa . Aa = Ob . Bb. Oa : Ob = Bb : Aa.

Similarly Hence That is, b

D

B C O

A

a

But, by similar triangles,

Therefore

BD : Bb = Oa : Ob = Aa : AD. Oa : Bb = Bb : Aa = Aa : OB,

that is, Bb and Oa are the two mean proportionals between OA and OB. It is impossible to construct the circle whose centre is C by Euclidean geometry, but Apollonius gave a mechanical way of describing it. This construction is quoted by several Arabic writers. In one of the most brilliant passages of his Aper¸cu historique Chasles remarks that, while Archimedes and Apollonius were the most able geometricians of the old world, their works are distinguished by a contrast which runs through the whole subsequent history of geometry. Archimedes, in attacking the problem of the quadrature of curvilinear areas, established the principles of the geometry which rests on measurements; this naturally gave rise to the infinitesimal calculus, and in fact the method of exhaustions as used by Archimedes does not differ in principle from the method of limits as used by Newton. Apollonius, on the other hand, in investigating the properties of conic sections by

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means of transversals involving the ratio of rectilineal distances and of perspective, laid the foundations of the geometry of form and position. Eratosthenes.1 Among the contemporaries of Archimedes and Apollonius I may mention Eratosthenes. Born at Cyrene in 275 b.c., he was educated at Alexandria—perhaps at the same time as Archimedes, of whom he was a personal friend—and Athens, and was at an early age entrusted with the care of the university library at Alexandria, a post which probably he occupied till his death. He was the Admirable Crichton of his age, and distinguished for his athletic, literary, and scientific attainments: he was also something of a poet. He lost his sight by ophthalmia, then as now a curse of the valley of the Nile, and, refusing to live when he was no longer able to read, he committed suicide in 194 b.c. In science he was chiefly interested in astronomy and geodesy, and he constructed various astronomical instruments which were used for some centuries at the university. He suggested the calendar (now known as Julian), in which every fourth year contains 366 days; and he determined the obliquity of the ecliptic as 23◦ 510 2000 . He measured the length of a degree on the earth’s surface, making it to be about 79 miles, which is too long by nearly 10 miles, and thence calculated the circumference of the earth to be 252,000 stadia. If we take the Olympic stadium of 202 14 yards, this is equivalent to saying that the radius is about 4600 miles, but there was also an Egyptian stadium, and if he used this he estimated the radius as 3925 miles, which is very near the truth. The principle used in the determination is correct. Of Eratosthenes’s work in mathematics we have two extant illustrations: one in a description of an instrument to duplicate a cube, and the other in a rule he gave for constructing a table of prime numbers. The former is given in many books. The latter, called the “sieve of Eratosthenes,” was as follows: write down all the numbers from 1 upwards; then every second number from 2 is a multiple of 2 and may be cancelled; every third number from 3 is a multiple of 3 and may be cancelled; every fifth number from 5 is a multiple of 5 and may be cancelled; and so on. It has been estimated that it would involve working for about 300 hours to thus find the primes in the numbers from 1 to 1,000,000. The labour of determining whether any particular number 1

The works of Eratosthenes exist only in fragments. A collection of these was published by G. Bernhardy at Berlin in 1822: some additional fragments were printed by E. Hillier, Leipzig, 1872.

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is a prime may be, however, much shortened by observing that if a number can be expressed as the product of two factors, one must be less and the other greater than the square root of the number, unless the number is the square of a prime, in which case the two factors are equal. Hence every composite number must be divisible by a prime which is not greater than its square root. The second century before Christ. The third century before Christ, which opens with the career of Euclid and closes with the death of Apollonius, is the most brilliant era in the history of Greek mathematics. But the great mathematicians of that century were geometricians, and under their influence attention was directed almost solely to that branch of mathematics. With the methods they used, and to which their successors were by tradition confined, it was hardly possible to make any further great advance: to fill up a few details in a work that was completed in its essential parts was all that could be effected. It was not till after the lapse of nearly 1800 years that the genius of Descartes opened the way to any further progress in geometry, and I therefore pass over the numerous writers who followed Apollonius with but slight mention. Indeed it may be said roughly that during the next thousand years Pappus was the sole geometrician of great original ability; and during this long period almost the only other pure mathematicians of exceptional genius were Hipparchus and Ptolemy, who laid the foundations of trigonometry, and Diophantus, who laid those of algebra. Early in the second century, circ. 180 b.c., we find the names of three mathematicians—Hypsicles, Nicomedes, and Diocles—who in their own day were famous. Hypsicles. The first of these was Hypsicles, who added a fourteenth book to Euclid’s Elements in which the regular solids were discussed. In another small work, entitled Risings, we find for the first time in Greek mathematics a right angle divided in the Babylonian manner into ninety degrees; possibly Eratosthenes may have previously estimated angles by the number of degrees they contain, but this is only a matter of conjecture. Nicomedes. The second was Nicomedes, who invented the curve known as the conchoid or the shell-shaped curve. If from a fixed point S a line be drawn cutting a given fixed straight line in Q, and if P be taken on SQ so that the length QP is constant (say d), then the locus of

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P is the conchoid. Its equation may be put in the form r = a sec θ ± d. It is easy with its aid to trisect a given angle or to duplicate a cube; and this no doubt was the cause of its invention. Diocles. The third of these mathematicians was Diocles, the inventor of the curve known as the cissoid or the ivy-shaped curve, which, like the conchoid, was used to give a solution of the duplication problem. He defined it thus: let AOA0 and BOB 0 be two fixed diameters of a circle at right angles to one another. Draw two chords QQ0 and RR0 parallel to BOB 0 and equidistant from it. Then the locus of the intersection of AR and QQ0 will be the cissoid. Its equation can be expressed in the form y 2 (2a − x) = x3 . The curve may be used to duplicate the cube. For, if OA and OE be the two lines between which it is required to insert two geometrical means, and if, in the figure constructed as above, A0 E cut the cissoid in P , and AP cut OB in D, we have OD3 = OA2 . OE. Thus OD is one of the means required, and the other mean can be found at once. Diocles also solved (by the aid of conic sections) a problem which had been proposed by Archimedes, namely, to draw a plane which will divide a sphere into two parts whose volumes shall bear to one another a given ratio. Perseus. Zenodorus. About a quarter of a century later, say about 150 b.c., Perseus investigated the various plane sections of the anchor-ring, and Zenodorus wrote a treatise on isoperimetrical figures. Part of the latter work has been preserved; one proposition which will serve to show the nature of the problems discussed is that “of segments of circles, having equal arcs, the semicircle is the greatest.” Towards the close of this century we find two mathematicians who, by turning their attention to new subjects, gave a fresh stimulus to the study of mathematics. These were Hipparchus and Hero. Hipparchus.1 Hipparchus was the most eminent of Greek astronomers—his chief predecessors being Eudoxus, Aristarchus, Archimedes, and Eratosthenes. Hipparchus is said to have been born about 160 b.c. at Nicaea in Bithynia; it is probable that he spent some years at Alexandria, but finally he took up his abode at Rhodes where he made most of his observations. Delambre has obtained an ingenious 1

See C. Manitius, Hipparchi in Arati et Eudoxi Phaenomena Commentarii, Leipzig, 1894, and J. B. J. Delambre, Histoire de l’astronomie ancienne, Paris, 1817, vol. i, pp. 106–189. S. P. Tannery in his Recherches sur l’histoire de l’astronomie ancienne, Paris, 1893, argues that the work of Hipparchus has been overrated, but I have adopted the view of the majority of writers on the subject.

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confirmation of the tradition which asserted that Hipparchus lived in the second century before Christ. Hipparchus in one place says that the longitude of a certain star η Canis observed by him was exactly 90◦ , and it should be noted that he was an extremely careful observer. Now in 1750 it was 116◦ 40 1000 , and, as the first point of Aries regredes at the rate of 50.200 a year, the observation was made about 120 b.c. Except for a short commentary on a poem of Aratus dealing with astronomy all his works are lost, but Ptolemy’s great treatise, the Almagest, described below, was founded on the observations and writings of Hipparchus, and from the notes there given we infer that the chief discoveries of Hipparchus were as follows. He determined the duration of the year to within six minutes of its true value. He calculated the inclination of the ecliptic and equator as 23◦ 510 ; it was actually at that time 23◦ 460 . He estimated the annual precession of the equinoxes as 5900 ; it is 50.200 . He stated the lunar parallax as 570 , which is nearly correct. He worked out the eccentricity of the solar orbit as 1/24; it is very approximately 1/30. He determined the perigee and mean motion of the sun and of the moon, and he calculated the extent of the shifting of the plane of the moon’s motion. Finally he obtained the synodic periods of the five planets then known. I leave the details of his observations and calculations to writers who deal specially with astronomy such as Delambre; but it may be fairly said that this work placed the subject for the first time on a scientific basis. To account for the lunar motion Hipparchus supposed the moon to move with uniform velocity in a circle, the earth occupying a position near (but not at) the centre of this circle. This is equivalent to saying that the orbit is an epicycle of the first order. The longitude of the moon obtained on this hypothesis is correct to the first order of small quantities for a few revolutions. To make it correct for any length of time Hipparchus further supposed that the apse line moved forward about 3◦ a month, thus giving a correction for eviction. He explained the motion of the sun in a similar manner. This theory accounted for all the facts which could be determined with the instruments then in use, and in particular enabled him to calculate the details of eclipses with considerable accuracy. He commenced a series of planetary observations to enable his successors to frame a theory to account for their motions; and with great perspicacity he predicted that to do this it would be necessary to introduce epicycles of a higher order, that is, to introduce three or more circles the centre of each successive one moving uniformly along the

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circumference of the preceding one. He also formed a list of 1080 of the fixed stars. It is said that the sudden appearance in the heavens of a new and brilliant star called his attention to the need of such a catalogue; and the appearance of such a star during his lifetime is confirmed by Chinese records. No further advance in the theory of astronomy was made until the time of Copernicus, though the principles laid down by Hipparchus were extended and worked out in detail by Ptolemy. Investigations such as these naturally led to trigonometry, and Hipparchus must be credited with the invention of that subject. It is known that in plane trigonometry he constructed a table of chords of arcs, which is practically the same as one of natural sines; and that in spherical trigonometry he had some method of solving triangles: but his works are lost, and we can give no details. It is believed, however, that the elegant theorem, printed as Euc. vi, d, and generally known as Ptolemy’s Theorem, is due to Hipparchus and was copied from him by Ptolemy. It contains implicitly the addition formulae for sin(A ± B) and cos(A ± B); and Carnot showed how the whole of elementary plane trigonometry could be deduced from it. I ought also to add that Hipparchus was the first to indicate the position of a place on the earth by means of its latitude and longitude. Hero.1 The second of these mathematicians was Hero of Alexandria, who placed engineering and land-surveying on a scientific basis. He was a pupil of Ctesibus, who invented several ingenious machines, and is alluded to as if he were a mathematician of note. It is not likely that Hero flourished before 80 b.c., but the precise period at which he lived is uncertain. In pure mathematics Hero’s principal and most characteristic work consists of (i) some elementary geometry, with applications to the determination of the areas of fields of given shapes; (ii) propositions on finding the volumes of certain solids, with applications to theatres, 1

See Recherches sur la vie et les ouvrages d’H´eron d’Alexandrie by T. H. Martin in vol. iv of M´emoires pr´esent´es . . . ` a l’acad´emie d’inscriptions, Paris, 1854; see also Loria, book iii, chap. v, pp. 107–128, and Cantor, chaps. xviii, xix. On the work entitled Definitions, which is attributed to Hero, see S. P. Tannery, chaps. xiii, xiv, and an article by G. Friedlein in Boncompagni’s Bulletino di bibliografia March 1871, vol. iv, pp. 93–126. Editions of the extant works of Hero were published in Teubner’s series, Leipzig, 1899, 1900, 1903. An English translation of the Πνευματικά was published by B. Woodcroft and J. G. Greenwood, London, 1851: drawings of the apparatus are inserted.

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baths, banquet-halls, and so on; (iii) a rule to find the height of an inaccessible object; and (iv) tables of weights and measures. He invented a solution of the duplication problem which is practically the same as that which Apollonius had already discovered. Some commentators think that he knew how to solve a quadratic equation even when the coefficients were not numerical; but this is doubtful. He proved the formula that the area of a triangle is equal to {s(s − a)(s − b)(s − c)}1/2 , where s is the semiperimeter, and a, b, c, the lengths of the sides, and gave as an illustration a triangle whose sides were in the ratio 13:14:15. He seems to have been acquainted with the trigonometry of Hipparchus, and the values of cot 2π/n are computed for various values of n, but he nowhere quotes a formula or expressly uses the value of the sine; it is probable that like the later Greeks he regarded trigonometry as forming an introduction to, and being an integral part of, astronomy. A

E

F O

B

D

L

C

H

K

The following is the manner in which he solved1 the problem to 1

In his Dioptra, Hultsch, part viii, pp. 235–237. It should be stated that some

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find the area of a triangle ABC the length of whose sides are a, b, c. Let s be the semiperimeter of the triangle. Let the inscribed circle touch the sides in D, E, F , and let O be its centre. On BC produced take H so that CH = AF , therefore BH = s. Draw OK at right angles to OB, and CK at right angles to BC; let them meet in K. The area ABC or 4 is equal to the sum of the areas OBC, OCA, OAB = 12 ar + 21 br + 12 cr = sr, that is, is equal to BH . OD. He then shews that the angle OAF = angle CBK; hence the triangles OAF and CBK are similar. ∴ BC : CK = AF : OF = CH : OD, ∴ BC : CH = CK : OD = CL : LD, ∴ BH : CH = CD : LD, ∴ BH 2 : CH . BH = CD . BD : LD . BD = CD . BD : OD2 . Hence 1

1

4 = BH . OD = {CH . BH . CD . BD} 2 = {(s − a)s(s − c)(s − b)} 2 . In applied mathematics Hero discussed the centre of gravity, the five simple machines, and the problem of moving a given weight with a given power; and in one place he suggested a way in which the power of a catapult could be tripled. He also wrote on the theory of hydraulic machines. He described a theodolite and cyclometer, and pointed out various problems in surveying for which they would be useful. But the most interesting of his smaller works are his Πνευματικά and Αὐτόματα, containing descriptions of about 100 small machines and mechanical toys, many of which are ingenious. In the former there is an account of a small stationary steam-engine which is of the form now known as Avery’s patent: it was in common use in Scotland at the beginning of this century, but is not so economical as the form introduced by Watt. There is also an account of a double forcing pump to be used as a fire-engine. It is probable that in the hands of Hero these instruments never got beyond models. It is only recently that general attention has been directed to his discoveries, though Arago had alluded to them in his ´eloge on Watt. All this is very different from the classical geometry and arithmetic of Euclid, or the mechanics of Archimedes. Hero did nothing to extend a knowledge of abstract mathematics; he learnt all that the text-books critics think that this is an interpolation, and is not due to Hero.

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of the day could teach him, but he was interested in science only on account of its practical applications, and so long as his results were true he cared nothing for the logical accuracy of the process by which he arrived at them. Thus, in finding the area of a triangle, he took the square root of the product of four lines. The classical Greek geometricians permitted the use of the square and the cube of a line because these could be represented geometrically, but a figure of four dimensions is inconceivable, and certainly they would have rejected a proof which involved such a conception. The first century before Christ. The successors of Hipparchus and Hero did not avail themselves of the opportunity thus opened of investigating new subjects, but fell back on the well-worn subject of geometry. Amongst the more eminent of these later geometricians were Theodosius and Dionysodorus, both of whom flourished about 50 b.c. Theodosius. Theodosius was the author of a complete treatise on the geometry of the sphere, and of two works on astronomy.1 Dionysodorus. Dionysodorus is known to us only by his solution2 of the problem to divide a hemisphere by a plane parallel to its base into two parts, whose volumes shall be in a given ratio. Like the solution by Diocles of the similar problem for a sphere above alluded to, it was effected by the aid of conic sections. Pliny says that Dionysodorus determined the length of the radius of the earth approximately as 42,000 stadia, which, if we take the Olympic stadium of 202 14 yards, is a little less than 5000 miles; we do not know how it was obtained. This may be compared with the result given by Eratosthenes and mentioned above. End of the First Alexandrian School. The administration of Egypt was definitely undertaken by Rome in 30 b.c. The closing years of the dynasty of the Ptolemies and the earlier years of the Roman occupation of the country were marked by much disorder, civil and political. The studies of the university were 1

The work on the sphere was edited by I. Barrow, Cambridge, 1675, and by E. Nizze, Berlin, 1852. The works on astronomy were published by Dasypodius in 1572. 2 It is reproduced in H. Suter’s Geschichte der mathematischen Wissenschaften, second edition, Z¨ urich, 1873, p. 101.

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naturally interrupted, and it is customary to take this time as the close of the first Alexandrian school.

78

CHAPTER V. the second alexandrian school.1 30 b.c.–641 a.d. I concluded the last chapter by stating that the first school of Alexandria may be said to have come to an end at about the same time as the country lost its nominal independence. But, although the schools at Alexandria suffered from the disturbances which affected the whole Roman world in the transition, in fact if not in name, from a republic to an empire, there was no break of continuity; the teaching in the university was never abandoned; and as soon as order was again established, students began once more to flock to Alexandria. This time of confusion was, however, contemporaneous with a change in the prevalent views of philosophy which thenceforward were mostly neo-platonic or neo-pythagorean, and it therefore fitly marks the commencement of a new period. These mystical opinions reacted on the mathematical school, and this may partially account for the paucity of good work. Though Greek influence was still predominant and the Greek language always used, Alexandria now became the intellectual centre for most of the Mediterranean nations which were subject to Rome. It should be added, however, that the direct connection with it of many of the mathematicians of this time is at least doubtful, but their knowledge was ultimately obtained from the Alexandrian teachers, and they are usually described as of the second Alexandrian school. Such mathematics as were taught at Rome were derived from Greek sources, and we may therefore conveniently consider their extent in connection with this chapter. 1

For authorities, see footnote above on p. 41. All dates given hereafter are to be taken as anno domini unless the contrary is expressly stated.

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The first century after Christ. There is no doubt that throughout the first century after Christ geometry continued to be that subject in science to which most attention was devoted. But by this time it was evident that the geometry of Archimedes and Apollonius was not capable of much further extension; and such geometrical treatises as were produced consisted mostly of commentaries on the writings of the great mathematicians of a preceding age. In this century the only original works of any ability of which we know anything were two by Serenus and one by Menelaus. Serenus. Menelaus. Those by Serenus of Antissa or of Antinoe, circ. 70, are on the plane sections of the cone and cylinder,1 in the course of which he lays down the fundamental proposition of transversals. That by Menelaus of Alexandria, circ. 98, is on spherical trigonometry, investigated in the Euclidean method.2 The fundamental theorem on which the subject is based is the relation between the six segments of the sides of a spherical triangle, formed by the arc of a great circle which cuts them [book iii, prop. 1]. Menelaus also wrote on the calculation of chords, that is, on plane trigonometry; this is lost. Nicomachus. Towards the close of this century, circ. 100, a Jew, Nicomachus, of Gerasa, published an Arithmetic,3 which (or rather the Latin translation of it) remained for a thousand years a standard authority on the subject. Geometrical demonstrations are here abandoned, and the work is a mere classification of the results then known, with numerical illustrations: the evidence for the truth of the propositions enunciated, for I cannot call them proofs, being in general an induction from numerical instances. The object of the book is the study of the properties of numbers, and particularly of their ratios. Nicomachus commences with the usual distinctions between even, odd, prime, and perfect numbers; he next discusses fractions in a somewhat clumsy manner; he then turns to polygonal and to solid numbers; and finally treats of ratio, proportion, and the progressions. Arithmetic of this kind is usually termed Boethian, and the work of Boethius on it was a recognised text-book in the middle ages. 1

These have been edited by J. L. Heiberg, Leipzig, 1896; and by E. Halley, Oxford, 1710. 2 This was translated by E. Halley, Oxford, 1758. 3 The work has been edited by R. Hoche, Leipzig, 1866.

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The second century after Christ. Theon. Another text-book on arithmetic on much the same lines as that of Nicomachus was produced by Theon of Smyrna, circ. 130. It formed the first book of his work1 on mathematics, written with the view of facilitating the study of Plato’s writings. Thymaridas. Another mathematician, reckoned by some writers as of about the same date as Theon, was Thymaridas, who is worthy of notice from the fact that he is the earliest known writer who explicitly enunciates an algebraical theorem. He states that, if the sum of any number of quantities be given, and also the sum of every pair which contains one of them, then this quantity is equal to one (n − 2)th part of the difference between the sum of these pairs and the first given sum. Thus, if

and if then

x1 + x2 + . . . + xn = S, x1 + x2 = s2 , x1 + x3 = s3 , . . . , and x1 + xn = sn , x1 = (s2 + s3 + . . . + sn − S)/(n − 2).

He does not seem to have used a symbol to denote the unknown quantity, but he always represents it by the same word, which is an approximation to symbolism. Ptolemy.2 About the same time as these writers Ptolemy of Alexandria, who died in 168, produced his great work on astronomy, which will preserve his name as long as the history of science endures. This treatise is usually known as the Almagest: the name is derived from the Arabic title al midschisti, which is said to be a corruption of μεγίστη [μαθηματική] σύνταξις. The work is founded on the writings of Hipparchus, and, though it did not sensibly advance the theory of the subject, it presents the views of the older writer with a completeness and elegance which will always make it a standard treatise. We gather from it that Ptolemy made observations at Alexandria from the years 1

The Greek text of those parts which are now extant, with a French translation, was issued by J. Dupuis, Paris, 1892. 2 See the article Ptolemaeus Claudius, by A. De Morgan in Smith’s Dictionary of Greek and Roman Biography, London, 1849; S. P. Tannery, Recherches sur l’histoire de l’astronomie ancienne, Paris, 1893; and J. B. J. Delambre, Histoire de l’astronomie ancienne, Paris, 1817, vol. ii. An edition of all the works of Ptolemy which are now extant was published at Bˆ ale in 1551. The Almagest with various minor works was edited by M. Halma, 12 vols. Paris, 1813–28, and a new edition, in two volumes, by J. L. Heiberg, Leipzig, 1898, 1903, 1907.

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125 to 150; he, however, was but an indifferent practical astronomer, and the observations of Hipparchus are generally more accurate than those of his expounder. The work is divided into thirteen books. In the first book Ptolemy discusses various preliminary matters; treats of trigonometry, plane or spherical; gives a table of chords, that is, of natural sines (which is substantially correct and is probably taken from the lost work of Hipparchus); and explains the obliquity of the ecliptic; in this book he uses degrees, minutes, and seconds as measures of angles. The second book is devoted chiefly to phenomena depending on the spherical form of the earth: he remarks that the explanations would be much simplified if the earth were supposed to rotate on its axis once a day, but states that this hypothesis is inconsistent with known facts. In the third book he explains the motion of the sun round the earth by means of excentrics and epicycles: and in the fourth and fifth books he treats the motion of the moon in a similar way. The sixth book is devoted 17 , as to the theory of eclipses; and in it he gives 3◦ 80 3000 , that is 3 120 the approximate value of π, which is equivalent to taking it equal to 3.1416. The seventh and eighth books contain a catalogue (probably copied from Hipparchus) of 1028 fixed stars determined by indicating those, three or more, that appear to be in a plane passing through the observer’s eye: and in another work Ptolemy added a list of annual sidereal phenomena. The remaining books are given up to the theory of the planets. This work is a splendid testimony to the ability of its author. It became at once the standard authority on astronomy, and remained so till Copernicus and Kepler shewed that the sun and not the earth must be regarded as the centre of the solar system. The idea of excentrics and epicycles on which the theories of Hipparchus and Ptolemy are based has been often ridiculed in modern times. No doubt at a later time, when more accurate observations had been made, the necessity of introducing epicycle on epicycle in order to bring the theory into accordance with the facts made it very complicated. But De Morgan has acutely observed that in so far as the ancient astronomers supposed that it was necessary to resolve every celestial motion into a series of uniform circular motions they erred greatly, but that, if the hypothesis be regarded as a convenient way of expressing known facts, it is not only legitimate but convenient. The theory suffices to describe either the angular motion of the heavenly bodies or their change in distance. The ancient astronomers were concerned only

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with the former question, and it fairly met their needs; for the latter question it is less convenient. In fact it was as good a theory as for their purposes and with their instruments and knowledge it was possible to frame, and corresponds to the expression of a given function as a sum of sines or cosines, a method which is of frequent use in modern analysis. In spite of the trouble taken by Delambre it is almost impossible to separate the results due to Hipparchus from those due to Ptolemy. But Delambre and De Morgan agree in thinking that the observations quoted, the fundamental ideas, and the explanation of the apparent solar motion are due to Hipparchus; while all the detailed explanations and calculations of the lunar and planetary motions are due to Ptolemy. E A

F

B

N

M C

G

D

H The Almagest shews that Ptolemy was a geometrician of the first rank, though it is with the application of geometry to astronomy that he is chiefly concerned. He was also the author of numerous other treatises. Amongst these is one on pure geometry in which he proposed to cancel Euclid’s postulate on parallel lines, and to prove it in the following manner. Let the straight line EF GH meet the two straight lines AB and CD so as to make the sum of the angles BF G and F GD equal to two right angles. It is required to prove that AB and CD are parallel. If possible let them not be parallel, then they will meet when produced say at M (or N ). But the angle AF G is the supplement of BF G, and is therefore equal to F GD: similarly the angle F GC is equal to the angle BF G. Hence the sum of the angles AF G and F GC is equal to two right angles, and the lines BA and DC will therefore if produced meet at N (or M ). But two straight lines cannot enclose a space, therefore AB and CD cannot meet when produced, that is, they are parallel. Conversely, if AB and CD be parallel, then AF and CG are not less parallel than F B and GD; and therefore whatever be the sum of the angles AF G and F GC such also must be the sum of

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the angles F GD and BF G. But the sum of the four angles is equal to four right angles, and therefore the sum of the angles BF G and F GD must be equal to two right angles. Ptolemy wrote another work to shew that there could not be more than three dimensions in space: he also discussed orthographic and stereographic projections with special reference to the construction of sun-dials. He wrote on geography, and stated that the length of one degree of latitude is 500 stadia. A book on sound is sometimes attributed to him, but on doubtful authority. The third century after Christ. Pappus. Ptolemy had shewn not only that geometry could be applied to astronomy, but had indicated how new methods of analysis like trigonometry might be thence developed. He found however no successors to take up the work he had commenced so brilliantly, and we must look forward 150 years before we find another geometrician of any eminence. That geometrician was Pappus who lived and taught at Alexandria about the end of the third century. We know that he had numerous pupils, and it is probable that he temporarily revived an interest in the study of geometry. Pappus wrote several books, but the only one which has come down to us is his Συναγωγή,1 a collection of mathematical papers arranged in eight books of which the first and part of the second have been lost. This collection was intended to be a synopsis of Greek mathematics together with comments and additional propositions by the editor. A careful comparison of various extant works with the account given of them in this book shews that it is trustworthy, and we rely largely on it for our knowledge of other works now lost. It is not arranged chronologically, but all the treatises on the same subject are grouped together, and it is most likely that it gives roughly the order in which the classical authors were read at Alexandria. Probably the first book, which is now lost, was on arithmetic. The next four books deal with geometry exclusive of conic sections; the sixth with astronomy including, as subsidiary subjects, optics and trigonometry; the seventh with analysis, conics, and porisms; and the eighth with mechanics. The last two books contain a good deal of original work by Pappus; at the same time it should be remarked that in two or three cases he 1

It has been published by F. Hultsch, Berlin, 1876–8.

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has been detected in appropriating proofs from earlier authors, and it is possible he may have done this in other cases. Subject to this suspicion we may say that Pappus’s best work is in geometry. He discovered the directrix in the conic sections, but he investigated only a few isolated properties: the earliest comprehensive account was given by Newton and Boscovich. As an illustration of his power I may mention that he solved [book vii, prop. 107] the problem to inscribe in a given circle a triangle whose sides produced shall pass through three collinear points. This question was in the eighteenth century generalised by Cramer by supposing the three given points to be anywhere; and was considered a difficult problem.1 It was sent in 1742 as a challenge to Castillon, and in 1776 he published a solution. Lagrange, Euler, Lhulier, Fuss, and Lexell also gave solutions in 1780. A few years later the problem was set to a Neapolitan lad A. Giordano, who was only 16 but who had shewn marked mathematical ability, and he extended it to the case of a polygon of n sides which pass through n given points, and gave a solution both simple and elegant. Poncelet extended it to conics of any species and subject to other restrictions. In mechanics Pappus shewed that the centre of mass of a triangular lamina is the same as that of an inscribed triangular lamina whose vertices divide each of the sides of the original triangle in the same ratio. He also discovered the two theorems on the surface and volume of a solid of revolution which are still quoted in text-books under his name: these are that the volume generated by the revolution of a curve about an axis is equal to the product of the area of the curve and the length of the path described by its centre of mass; and the surface is equal to the product of the perimeter of the curve and the length of the path described by its centre of mass. The problems above mentioned are but samples of many brilliant but isolated theorems which were enunciated by Pappus. His work as a whole and his comments shew that he was a geometrician of power; but it was his misfortune to live at a time when but little interest was taken in geometry, and when the subject, as then treated, had been practically exhausted. Possibly a small tract2 on multiplication and division of sexagesimal 1

For references to this problem see a note by H. Brocard in L’Interm´ediaire des math´ematiciens, Paris, 1904, vol. xi, pp. 219–220. 2 It was edited by C. Henry, Halle, 1879, and is valuable as an illustration of practical Greek arithmetic.

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fractions, which would seem to have been written about this time, is due to Pappus. The fourth century after Christ. Throughout the second and third centuries, that is, from the time of Nicomachus, interest in geometry had steadily decreased, and more and more attention had been paid to the theory of numbers, though the results were in no way commensurate with the time devoted to the subject. It will be remembered that Euclid used lines as symbols for any magnitudes, and investigated a number of theorems about numbers in a strictly scientific manner, but he confined himself to cases where a geometrical representation was possible. There are indications in the works of Archimedes that he was prepared to carry the subject much further: he introduced numbers into his geometrical discussions and divided lines by lines, but he was fully occupied by other researches and had no time to devote to arithmetic. Hero abandoned the geometrical representation of numbers, but he, Nicomachus, and other later writers on arithmetic did not succeed in creating any other symbolism for numbers in general, and thus when they enunciated a theorem they were content to verify it by a large number of numerical examples. They doubtless knew how to solve a quadratic equation with numerical coefficients—for, as pointed out above, geometrical solutions of the equations ax2 − bx + c = 0 and ax2 + bx − c = 0 are given in Euc. vi, 28 and 29—but probably this represented their highest attainment. It would seem then that, in spite of the time given to their study, arithmetic and algebra had not made any sensible advance since the time of Archimedes. The problems of this kind which excited most interest in the third century may be illustrated from a collection of questions, printed in the Palatine Anthology, which was made by Metrodorus at the beginning of the next century, about 310. Some of them are due to the editor, but some are of an anterior date, and they fairly illustrate the way in which arithmetic was leading up to algebraical methods. The following are typical examples. “Four pipes discharge into a cistern: one fills it in one day; another in two days; the third in three days; the fourth in four days: if all run together how soon will they fill the cistern?” “Demochares has lived a fourth of his life as a boy; a fifth as a youth; a third as a man; and has spent thirteen years in his dotage: how old is he?” “Make a crown of gold, copper, tin, and iron weighing 60 minae: gold and copper shall be two-thirds of it; gold

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and tin three-fourths of it; and gold and iron three-fifths of it: find the weights of the gold, copper, tin, and iron which are required.” The last is a numerical illustration of Thymaridas’s theorem quoted above. It is believed that these problems were solved by rhetorical algebra, that is, by a process of algebraical reasoning expressed in words and without the use of any symbols. This, according to Nesselmann, is the first stage in the development of algebra, and we find it used both by Ahmes and by the earliest Arabian, Persian, and Italian algebraists: examples of its use in the solution of a geometrical problem and in the rule for the solution of a quadratic equation are given later.1 On this view then a rhetorical algebra had been gradually evolved by the Greeks, or was then in process of evolution. Its development was however very imperfect. Hankel, who is no unfriendly critic, says that the results attained as the net outcome of the work of six centuries on the theory of numbers are, whether we look at the form or the substance, unimportant or even childish, and are not in any way the commencement of a science. In the midst of this decaying interest in geometry and these feeble attempts at algebraic arithmetic, a single algebraist of marked originality suddenly appeared who created what was practically a new science. This was Diophantus who introduced a system of abbreviations for those operations and quantities which constantly recur, though in using them he observed all the rules of grammatical syntax. The resulting science is called by Nesselmann syncopated algebra: it is a sort of shorthand. Broadly speaking, it may be said that European algebra did not advance beyond this stage until the close of the sixteenth century. Modern algebra has progressed one stage further and is entirely symbolic; that is, it has a language of its own and a system of notation which has no obvious connection with the things represented, while the operations are performed according to certain rules which are distinct from the laws of grammatical construction. Diophantus.2 All that we know of Diophantus is that he lived at Alexandria, and that most likely he was not a Greek. Even the date of his career is uncertain; it cannot reasonably be put before the middle of the third century, and it seems probable that he was alive in the early 1

See below, pp. 168, 174. A critical edition of the collected works of Diophantus was edited by S. P. Tannery, 2 vols., Leipzig, 1893; see also Diophantos of Alexandria, by T. L. Heath, Cambridge, 1885; and Loria, book v, chap. v, pp. 95–158. 2

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years of the fourth century, that is, shortly after the death of Pappus. He was 84 when he died. In the above sketch of the lines on which algebra has developed I credited Diophantus with the invention of syncopated algebra. This is a point on which opinions differ, and some writers believe that he only systematized the knowledge which was familiar to his contemporaries. In support of this latter opinion it may be stated that Cantor thinks that there are traces of the use of algebraic symbolism in Pappus, and are used Freidlein mentions a Greek papyrus in which the signs / and for addition and subtraction respectively; but no other direct evidence for the non-originality of Diophantus has been produced, and no ancient author gives any sanction to this opinion. Diophantus wrote a short essay on polygonal numbers; a treatise on algebra which has come down to us in a mutilated condition; and a work on porisms which is lost. The Polygonal Numbers contains ten propositions, and was probably his earliest work. In this he reverts to the classical system by which numbers are represented by lines, a construction is (if necessary) made, and a strictly deductive proof follows: it may be noticed that in it he quotes propositions, such as Euc. ii, 3, and ii, 8, as referring to numbers and not to magnitudes. His chief work is his Arithmetic. This is really a treatise on algebra; algebraic symbols are used, and the problems are treated analytically. Diophantus tacitly assumes, as is done in nearly all modern algebra, that the steps are reversible. He applies this algebra to find solutions (though frequently only particular ones) of several problems involving numbers. I propose to consider successively the notation, the methods of analysis employed, and the subject-matter of this work. First, as to the notation. Diophantus always employed a symbol to represent the unknown quantity in his equations, but as he had only one symbol he could not use more than one unknown at a time.1 The unknown quantity is called ὁ ἀριθμός, and is represented by 0 or o0 . It is usually printed as ς. In the plural it is denoted by ςς or ςς o`ι . This symbol may be a corruption of αρ , or perhaps it may be the final sigma of this word, or possibly it may stand for the word σωρός a heap.2 The square of the unknown is called δύναμις, and denoted by δ υ¯ : the cube 1

See, however, below, page 90, example (iii), for an instance of how he treated a problem involving two unknown quantities. 2 See above, page 4.

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κύβος, and denoted by κυ ; and so on up to the sixth power. The coefficients of the unknown quantity and its powers are numbers, and a numerical coefficient is written immediately after the quantity it multiplies: thus ς 0 α ¯ = x, and ςς oι ια = ςς ια = 11x. An absolute term is regarded as a certain number of units or μονάδες which are represented by µoˆ: thus µoˆα ¯ = 1, µoˆια = 11. There is no sign for addition beyond juxtaposition. Subtraction is represented by , and this symbol affects all the symbols that follow it. Equality is represented by ι. Thus ψ

κυˆ α ¯ ςς η¯ δ oˆ¯ µoˆα ¯ ι ςα ¯ (x3 + 8x) − (5x2 + 1) = x. ψ

represents

Diophantus also introduced a somewhat similar notation for fractions involving the unknown quantity, but into the details of this I need not here enter. It will be noticed that all these symbols are mere abbreviations for words, and Diophantus reasons out his proofs, writing these abbreviations in the middle of his text. In most manuscripts there is a marginal summary in which the symbols alone are used and which is really symbolic algebra; but probably this is the addition of some scribe of later times. This introduction of a contraction or a symbol instead of a word to represent an unknown quantity marks a greater advance than anyone not acquainted with the subject would imagine, and those who have never had the aid of some such abbreviated symbolism find it almost impossible to understand complicated algebraical processes. It is likely enough that it might have been introduced earlier, but for the unlucky system of numeration adopted by the Greeks by which they used all the letters of the alphabet to denote particular numbers and thus made it impossible to employ them to represent any number. Next, as to the knowledge of algebraic methods shewn in the book. Diophantus commences with some definitions which include an explanation of his notation, and in giving the symbol for minus he states that a subtraction multiplied by a subtraction gives an addition; by this he means that the product of −b and −d in the expansion of (a − b)(c − d) is +bd, but in applying the rule he always takes care that the numbers a, b, c, d are so chosen that a is greater than b and c is greater than d. The whole of the work itself, or at least as much as is now extant, is devoted to solving problems which lead to equations. It contains

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rules for solving a simple equation of the first degree and a binomial quadratic. Probably the rule for solving any quadratic equation was given in that part of the work which is now lost, but where the equation is of the form ax2 +bx+c = 0 he seems to have multiplied by a and then “completed the square” in much the same way as is now done: when the roots are negative or irrational the equation is rejected as “impossible,” and even when both roots are positive he never gives more than one, always taking the positive value of the square root. Diophantus solves one cubic equation, namely, x3 + x = 4x2 + 4 [book vi, prob. 19]. The greater part of the work is however given up to indeterminate equations between two or three variables. When the equation is between two variables, then, if it be of the first degree, he assumes a suitable value for one variable and solves the equation for the other. Most of his equations are of the form y 2 = Ax2 + Bx + C. Whenever A or C is equal to zero, he is able to solve the equation completely. When this is not the case, then, if A = a2 , he assumes y = ax + m; if C = c2 , he assumes y = mx + c; and lastly, if the equation can be put in the form y 2 = (ax ± b)2 + c2 , he assumes y = mx: where in each case m has some particular numerical value suitable to the problem under consideration. A few particular equations of a higher order occur, but in these he generally alters the problem so as to enable him to reduce the equation to one of the above forms. The simultaneous indeterminate equations involving three variables, or “double equations” as he calls them, which he considers are of the forms y 2 = Ax2 + Bx + C and z 2 = ax2 + bx + c. If A and a both vanish, he solves the equations in one of two ways. It will be enough to give one of his methods which is as follows: he subtracts and thus gets an equation of the form y 2 − z 2 = mx + n; hence, if y ± z = λ, then y ∓ z = (mx + n)/λ; and solving he finds y and z. His treatment of “double equations” of a higher order lacks generality and depends on the particular numerical conditions of the problem. Lastly, as to the matter of the book. The problems he attacks and the analysis he uses are so various that they cannot be described concisely and I have therefore selected five typical problems to illustrate his methods. What seems to strike his critics most is the ingenuity with which he selects as his unknown some quantity which leads to equations such as he can solve, and the artifices by which he finds numerical solutions of his equations. I select the following as characteristic examples. (i) Find four numbers, the sum of every arrangement three at a time

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being given; say 22, 24, 27, and 20 [book i, prob. 17]. Let x be the sum of all four numbers; hence the numbers are x − 22, x − 24, x − 27, and x − 20. ∴ x = (x − 22) + (x − 24) + (x − 27) + (x − 20). ∴ x = 31. ∴ the numbers are 9, 7, 4, and 11. (ii) Divide a number, such as 13 which is the sum of two squares 4 and 9, into two other squares [book ii, prob. 10]. He says that since the given squares are 22 and 32 he will take (x + 2)2 and (mx − 3)2 as the required squares, and will assume m = 2. ∴ (x + 2)2 + (2x − 3)2 = 13. ∴ x = 8/5. ∴ the required squares are 324/25 and 1/25. (iii) Find two squares such that the sum of the product and either is a square [book ii, prob. 29]. Let x2 and y 2 be the numbers. Then x2 y 2 + y 2 and x2 y 2 + x2 are squares. The first will be a square if x2 +1 be a square, which he assumes may be taken equal to (x − 2)2 , hence x = 3/4. He has now to make 9(y 2 + 1)/16 a square, to do this he assumes that 9y 2 + 9 = (3y − 4)2 , hence y = 7/24. Therefore the squares required are 9/16 and 49/576. It will be recollected that Diophantus had only one symbol for an unknown quantity; and in this example he begins by calling the unknowns x2 and 1, but as soon as he has found x he then replaces the 1 by the symbol for the unknown quantity, and finds it in its turn. (iv) To find a [rational ] right-angled triangle such that the line bisecting an acute angle is rational [book vi, prob. 18]. A

B

D

C

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His solution is as follows. Let ABC be the triangle of which C is the right-angle. Let the bisector AD = 5x, and let DC = 3x, hence AC = 4x. Next let BC be a multiple of 3, say 3, ∴ BD = 3−3x, hence AB = 4 − 4x (by Euc. vi, 3). Hence (4 − 4x)2 = 32 + (4x)2 (Euc. i, 47), ∴ x = 7/32. Multiplying by 32 we get for the sides of the triangle 28, 96, and 100; and for the bisector 35. (v) A man buys x measures of wine, some at 8 drachmae a measure, the rest at 5. He pays for them a square number of drachmae, such that, if 60 be added to it, the resulting number is x2 . Find the number he bought at each price [book v, prob. 33]. The price paid was x2 − 60, hence 8x > x2 − 60 and 5x < x2 − 60. From this it follows that x must be greater than 11 and less than 12. Again x2 − 60 is to be a square; suppose it is equal to (x − m)2 then x = (m2 + 60)/2m, we have therefore m2 + 60 < 12; 2m ∴ 19 < m < 21.

11

and < to represent greater than and less than. When he denoted the unknown quantity by a he represented a2 by aa, a3 by aaa, and so on. This is a distinct improvement on Vieta’s notation. The same symbolism was used by Wallis as late as 1685, but concurrently with the modern index notation which was introduced by Descartes. I need not allude to the other investigations of Harriot, as they are comparatively of small importance; extracts from some of them were published by S. P. Rigaud in 1833. Oughtred. Among those who contributed to the general adoption in England of these various improvements and additions to algorism and algebra was William Oughtred,1 who was born at Eton on March 5, 1574, and died at his vicarage of Albury in Surrey on June 30, 1660: it is sometimes said that the cause of his death was the excitement and delight which he experienced “at hearing the House of Commons [or Convention] had voted the King’s return”; a recent critic adds that it should be remembered “by way of excuse that he [Oughtred] was then eighty-six years old,” but perhaps the story is sufficiently discredited by the date of his death. Oughtred was educated at Eton and King’s College, Cambridge, of the latter of which colleges he was a fellow and for some time mathematical lecturer. His Clavis Mathematicae published in 1631 is a good systematic text-book on arithmetic, and it contains practically all that was then known on the subject. In this work he introduced the symbol × for multiplication. He also introduced the symbol : : in proportion: previously to his time a proportion such as a : b = c : d was usually written 1

See William Oughtred, by F. Cajori, Chicago, 1916. A complete edition of Oughtred’s works was published at Oxford in 1677.

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as a − b − c − d; he denoted it by a . b : : c . d. Wallis says that some found fault with the book on account of the style, but that they only displayed their own incompetence, for Oughtred’s “words be always full but not redundant.” Pell makes a somewhat similar remark. Oughtred also wrote a treatise on trigonometry published in 1657, in which abbreviations for sine, cosine, &c., were employed. This was really an important advance, but the works of Girard and Oughtred, in which they were used, were neglected and soon forgotten, and it was not until Euler reintroduced contractions for the trigonometrical functions that they were generally adopted. In this work the colon (i.e. the symbol :) was used to denote a ratio. We may say roughly that henceforth elementary arithmetic, algebra, and trigonometry were treated in a manner which is not substantially different from that now in use; and that the subsequent improvements introduced were additions to the subjects as then known, and not a rearrangement of them on new foundations. The origin of the more common symbols in algebra. It may be convenient if I collect here in one place the scattered remarks I have made on the introduction of the various symbols for the more common operations in algebra.1 The later Greeks, the Hindoos, and Jordanus indicated addition by mere juxtaposition. It will be observed that this is still the custom in arithmetic, where, for instance, 2 12 stands for 2 + 21 . The Italian algebraists, when they gave up expressing every operation in words at full length and introduced syncopated algebra, usually denoted plus by its initial letter P or p, a line being sometimes drawn through the letter to show that it was a contraction, or a symbol of operation, and not a quantity. The practice, however, was not uniform; Pacioli, for example, sometimes denoted plus by p¯, and sometimes by e, and Tartaglia commonly denoted it by φ. The German and English algebraists, on the other hand, introduced the sign + almost as soon as they used algorism, but they spoke of it as signum additorum and employed it only to indicate excess; they also used it with a special meaning in solutions by the method of false assumption. Widman used it as an abbreviation 1

See also two articles by C. Henry in the June and July numbers of the Revue Arch´eologique, 1879, vol. xxxvii, pp. 324–333, vol. xxxviii, pp. 1–10.

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for excess in 1489: by 1630 it was part of the recognised notation of algebra, and was used as a symbol of operation. Subtraction was indicated by Diophantus by an inverted and truncated ψ. The Hindoos denoted it by a dot. The Italian algebraists when they introduced syncopated algebra generally denoted minus by M or m, a line being sometimes drawn through the letter; but the practice was not uniform—Pacioli, for example, denoting it sometimes by m, ¯ and sometimes by de for demptus. The German and English algebraists introduced the present symbol which they described as signum subtractorum. It is most likely that the vertical bar in the symbol for plus was superimposed on the symbol for minus to distinguish the two. It may be noticed that Pacioli and Tartaglia found the sign − already used to denote a division, a ratio, or a proportion indifferently. The present sign for minus was in general use by about the year 1630, and was then employed as a symbol of operation. Vieta, Schooten, and others among their contemporaries employed the sign = written between two quantities to denote the difference between them; thus a = b means with them what we denote by a ∼ b. On the other hand, Barrow wrote −−: for the same purpose. I am not aware when or by whom the current symbol ∼ was first used with this signification. Oughtred in 1631 used the sign × to indicate multiplication; Harriot in 1631 denoted the operation by a dot; Descartes in 1637 indicated it by juxtaposition. I am not aware of any symbols for it which were in previous use. Leibnitz in 1686 employed the sign _ to denote multiplication. Division was ordinarily denoted by the Arab way of writing the quantities in the form of a fraction by means of a line drawn between a them in any of the forms a − b, a/b, or . Oughtred in 1631 employed b a dot to denote either division or a ratio. Leibnitz in 1686 employed the sign ^ to denote division. The colon (or symbol :), used to denote a ratio, occurs on the last two pages of Oughtred’s Canones Sinuum, published in 1657. I believe that the current symbol for division ÷ is only a combination of the − and the symbol : for a ratio; it was used by Johann Heinrich Rahn at Z¨ urich in 1659, and by John Pell in London in 1668. The symbol ÷÷ was used by Barrow and other writers of his time to indicate continued proportion. The current symbol for equality was introduced by Record in 1557; Xylander in 1575 denoted it by two parallel vertical lines; but in general

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8

8

till the year 1600 the word was written at length; and from then until the time of Newton, say about 1680, it was more frequently represented by or by than by any other symbol. Either of these latter signs was used as a contraction for the first two letters of the word aequalis. The symbol : : to denote proportion, or the equality of two ratios, was introduced by Oughtred in 1631, and was brought into common use by Wallis in 1686. There is no object in having a symbol to indicate the equality of two ratios which is different from that used to indicate the equality of other things, and it is better to replace it by the sign =. The sign > for is greater than and the sign < for is less than were introduced by Harriot in 1631, but Oughtred simultaneously invented and for the same purpose; and these latter were the symbols frequently used till the beginning of the eighteenth century, ex. gr. by Barrow. The symbols =| for is not equal to, > | for is not greater than, and

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